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Proposition 12.1.1. A quasi-ordering \( \leq \) on \( X \) is a well-quasi-ordering if and only if \( X \) contains neither an infinite antichain nor an infinite strictly decreasing sequence \( {x}_{0} > {x}_{1} > \ldots \) | Proof. The forward implication is trivial. Conversely, let \( {x}_{0},{x}_{1},\ldots \) be any infinite sequence in \( X \) . Let \( K \) be the complete graph on \( \mathbb{N} = \) \( \{ 0,1,\ldots \} \) . Colour the edges \( {ij}\left( {i < j}\right) \) of \( K \) with three colours: green if \( {x}_{i} \leq {x}_{j} ... | Yes |
Lemma 12.3.1. Let \( {t}_{1}{t}_{2} \) be any edge of \( T \) and let \( {T}_{1},{T}_{2} \) be the components of \( T - {t}_{1}{t}_{2} \), with \( {t}_{1} \in {T}_{1} \) and \( {t}_{2} \in {T}_{2} \) . Then \( {V}_{{t}_{1}} \cap {V}_{{t}_{2}} \) separates \( {U}_{1} \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{t \in ... | Proof. Both \( {t}_{1} \) and \( {t}_{2} \) lie on every \( t - {t}^{\prime } \) path in \( T \) with \( t \in {T}_{1} \) and \( {t}^{\prime } \in {T}_{2} \) . Therefore \( {U}_{1} \cap {U}_{2} \subseteq {V}_{{t}_{1}} \cap {V}_{{t}_{2}} \) by (T3), so all we have to show is that \( G \) has no edge \( {u}_{1}{u}_{2} \)... | Yes |
Lemma 12.3.3. Suppose that \( G \) is an \( {MH} \) with branch sets \( {U}_{h} \) , \( h \in V\left( H\right) \) . Let \( f : V\left( G\right) \rightarrow V\left( H\right) \) be the map assigning to each vertex of \( G \) the index of the branch set containing it. For all \( t \in T \) let \( {W}_{t} \mathrel{\text{:=... | Proof. The assertions (T1) and (T2) for \( \left( {T,\mathcal{W}}\right) \) follow immediately from the corresponding assertions for \( \left( {T,\mathcal{V}}\right) \) . Now let \( {t}_{1},{t}_{2},{t}_{3} \in T \) be as in (T3), and consider a vertex \( h \in {W}_{{t}_{1}} \cap {W}_{{t}_{3}} \) of \( H \) ; we show th... | Yes |
Lemma 12.3.4. Given a set \( W \subseteq V\left( G\right) \), there is either a \( t \in T \) such that \( W \subseteq {V}_{t} \), or there are vertices \( {w}_{1},{w}_{2} \in W \) and an edge \( {t}_{1}{t}_{2} \in T \) such that \( {w}_{1},{w}_{2} \) lie outside the set \( {V}_{{t}_{1}} \cap {V}_{{t}_{2}} \) and are s... | Proof. Let us orient the edges of \( T \) as follows. For each edge \( {t}_{1}{t}_{2} \in T \) , define \( {U}_{1},{U}_{2} \) as in Lemma 12.3.1; then \( {V}_{{t}_{1}} \cap {V}_{{t}_{2}} \) separates \( {U}_{1} \) from \( {U}_{2} \) . If \( {V}_{{t}_{1}} \cap {V}_{{t}_{2}} \) does not separate any two vertices of \( W ... | Yes |
Lemma 12.3.8. Any set of vertices separating two covers of a bramble also covers that bramble. | Proof. Since each set in the bramble is connected and meets both of the covers, it also meets any set separating these covers. | No |
Every graph \( G \) has a lean tree-decomposition of width \( \operatorname{tw}\left( G\right) \) . | There is now a short proof of Theorem 12.3.10; see the notes. The fact that this theorem gives us a useful property of minimum-width tree-decompositions 'for free' has made it a valuable tool wherever tree-decompositions are applied. | No |
Proposition 12.3.11. \( G \) is chordal if and only if \( G \) has a tree-decomposition into complete parts. | Proof. We apply induction on \( \left| G\right| \) . We first assume that \( G \) has a tree-decomposition \( \left( {T,\mathcal{V}}\right) \) such that \( G\left\lbrack {V}_{t}\right\rbrack \) is complete for every \( t \in T \) ; let us choose \( \left( {T,\mathcal{V}}\right) \) with \( \left| T\right| \) minimal. If... | Yes |
Corollary 12.3.12. \( \operatorname{tw}\left( G\right) = \min \{ \omega \left( H\right) - 1 \mid G \subseteq H;H \) chordal \( \} \) . | Proof. By Lemma 12.3.5 and Proposition 12.3.11, each of the graphs \( H \) considered for the minimum has a tree-decomposition of width \( \omega \left( H\right) - 1 \) . Every such tree-decomposition induced one of \( G \) by Lemma 12.3.2, so \( \operatorname{tw}\left( G\right) \leq \omega \left( H\right) - 1 \) for e... | Yes |
Proposition 12.4.1. A graph property \( \mathcal{P} \) can be expressed by forbidden minors if and only if it is closed under taking minors. | Proof. For the ’if’ part, note that \( \mathcal{P} = {\operatorname{Forb}}_{ \preccurlyeq }\left( \overline{\mathcal{P}}\right) \), where \( \overline{\mathcal{P}} \) is the \( \overline{\mathcal{P}} \) complement of \( \mathcal{P} \) . | No |
Proposition 12.4.2. A graph has tree-width \( < 3 \) if and only if it has no \( {K}^{4} \) minor. | Proof. By Lemma 12.3.5, we have \( \operatorname{tw}\left( {K}^{4}\right) \geq 3 \) . By Proposition 12.3.6, (12.3.5) therefore, a graph of tree-width \( < 3 \) cannot contain \( {K}^{4} \) as a minor. (12.3.6)\n\nConversely, let \( G \) be a graph without a \( {K}^{4} \) minor; we assume that (12.3.11) \( \left| G\rig... | Yes |
Given a graph \( H \), the graphs without an \( H \) minor have bounded tree-width if and only if \( H \) is planar. | To prove Theorem 12.4.3 we have to show that forbidding any planar graph \( H \) as a minor bounds the tree-width of a graph. In fact, we only have to show this for the special cases when \( H \) is a grid, because every planar graph is a minor of some grid. (To see this, take a drawing of the graph, fatten its vertice... | Yes |
Theorem 12.4.4. (Robertson & Seymour 1986)\n\nFor every integer \( r \) there is an integer \( k \) such that every graph of tree-width at least \( k \) has an \( r \times r \) grid minor. | Our proof of Theorem 12.4.4 proceeds as follows. Let \( r \) be given, and let \( G \) be any graph of large enough tree-width (depending on \( r \) ). We first show that \( G \) contains a large family \( \mathcal{A} = \left\{ {{A}_{1},\ldots ,{A}_{m}}\right\} \) of disjoint connected vertex sets such that each pair \... | Yes |
Lemma 12.4.6. Let \( k \geq 2 \) be an integer. Let \( T \) be a tree of maximum degree \( \leq 3 \) and \( X \subseteq V\left( T\right) \) . Then \( T \) has a set \( F \) of edges such that every component of \( T - F \) has between \( k \) and \( {2k} - 1 \) vertices in \( X \), except that one such component may ha... | Proof. We apply induction on \( \left| X\right| \) . If \( \left| X\right| \leq {2k} - 1 \) we put \( F = \varnothing \) . So assume that \( \left| X\right| \geq {2k} \) . Let \( e \) be an edge of \( T \) such that some component \( {T}^{\prime } \) of \( T - e \) has at least \( k \) vertices in \( X \) and \( \left|... | Yes |
Lemma 12.4.7. Let \( G \) be a bipartite graph with bipartition \( \{ A, B\} \) , \( \left| A\right| = a,\left| B\right| = b \), and let \( c \leq a \) and \( d \leq b \) be positive integers. Assume that \( G \) has at most \( \left( {a - c}\right) \left( {b - d}\right) /d \) edges. Then there exist \( C \subseteq A \... | Proof. As \( \parallel G\parallel \leq \left( {a - c}\right) \left( {b - d}\right) /d \), fewer than \( b - d \) vertices in \( B \) have more than \( \left( {a - c}\right) /d \) neighbours in \( A \) . Choose \( D \subseteq B \) so that \( \left| D\right| = d \) and each vertex in \( D \) has at most \( \left( {a - c}... | Yes |
Lemma 12.4.8. Every tree \( T \) of order at least \( r\left( {r - 1}\right) \) contains a good \( r \) -tuple of vertices. | Proof. Pick a vertex \( x \in T \) . Then \( T \) is the union of its subpaths \( {xTy} \) , where \( y \) ranges over its leaves. Hence unless one of these paths has at least \( r \) vertices, \( T \) has at least \( \left| T\right| /\left( {r - 1}\right) \geq r \) leaves. Since any path of \( r \) vertices and any se... | Yes |
For every \( n \in \mathbb{N} \) there exists a \( k \in \mathbb{N} \) such that every graph \( G \) not containing \( {K}^{n} \) as a minor has a tree-decomposition whose torsos are \( k \) -nearly embeddable in a surface in which \( {K}^{n} \) is not embeddable. | Note that there are only finitely many surfaces in which \( {K}^{n} \) is not embeddable. The set of those surfaces in the statement of Theorem 12.4.11 could therefore be replaced by just two surfaces: the orientable and the non-orientable surface of maximum genus in this set. Note also that the separators \( {V}_{t} \... | Yes |
Theorem 12.5.1. (Robertson & Seymour 1986-2004)\n\nThe finite graphs are well-quasi-ordered by the minor relation \( \\preccurlyeq \) . | We shall give a sketch of the proof of the graph minor theorem at the end of this section. | No |
Corollary 12.5.3. For every surface \( S \) there exists a finite set of graphs \( {H}_{1},\ldots ,{H}_{n} \) such that a graph is embeddable in \( S \) if and only if it contains none of \( {H}_{1},\ldots ,{H}_{n} \) as a minor. | The proof of Corollary 12.5.3 does not need the full strength of the minor theorem. We shall give a direct proof, which runs as follows. The main step is to prove that the graphs in \( {\mathcal{K}}_{\mathcal{P}\left( S\right) } \) do not contain arbitrarily large grids as minors (Lemma 12.5.4). Then their tree-width i... | No |
For every surface \( S \) there exists an integer \( r \) such that no graph that is minimal with the property of not being embeddable in \( S \) contains \( {H}^{r} \) as a topological minor. | Proof. Let \( G \) be a graph that cannot be embedded in \( S \) and is minimal with this property. Our proof will run roughly as follows. Since \( G \) is minimally not embeddable in \( S \), we can embed it in an only slightly larger surface \( {S}^{\prime } \). If \( G \) contains a very large \( {H}^{r} \) grid, th... | Yes |
Theorem 1.5. Let \( K \) be an oriented knot in (oriented) \( {S}^{3} \), and let \( X \) be its exterior. Then \( {H}_{1}\left( X\right) \) is canonically isomorphic to the integers \( \mathbb{Z} \) generated by the class of a simple closed curve \( \mu \) in \( \partial N \) that bounds a disc in \( N \) meeting \( K... | Proof. This result is true in any reasonable homology theory with integer coefficients; indeed, it follows at once from the relatively sophisticated theorem of Alexander duality. The following proof uses the Mayer-Vietoris theorem, which relates the homology of two spaces to that of their union and intersection. As it ... | Yes |
Theorem 1.7. Let \( L \) be an oriented link of \( n \) components in (oriented) \( {S}^{3} \) and let \( X \) be its exterior. Then \( {H}_{2}\left( X\right) = {\bigoplus }_{n - 1}\mathbb{Z} \) . Further, \( {H}_{1}\left( X\right) \) is canonically isomorphic to \( {\bigoplus }_{n}\mathbb{Z} \) generated by the homolo... | Proof. The proof of this is just an adaptation of that of the previous theorem. Here \( N \) is now a disjoint union of \( n \) solid tori. The map \( {H}_{3}\left( {S}^{3}\right) \rightarrow {H}_{2}\left( {X \cap N}\right) \) is the map \( \mathbb{Z} \rightarrow {\bigoplus }_{n}\mathbb{Z} \) that sends 1 to \( \left( ... | Yes |
Theorem 2.4. For any two knots \( {K}_{1} \) and \( {K}_{2} \) ,\n\n\[ g\left( {{K}_{1} + {K}_{2}}\right) = g\left( {K}_{1}\right) + g\left( {K}_{2}\right) . \] | Proof. Firstly, suppose that \( {K}_{1} \) and \( {K}_{2} \), together with minimal genus Seifert surfaces \( {F}_{1} \) and \( {F}_{2} \), are situated far apart in \( {S}^{3} \) . Each \( {F}_{i} \) is a connected surface with non-empty boundary, so elementary homology theory shows that \( {F}_{1} \cup {F}_{2} \) doe... | No |
Corollary 2.8. A knot can be expressed as a finite sum of prime knots. | Proof. If a knot is not prime, it can be expressed as the sum of two knots of smaller genus. Now use induction on the genus. | No |
Theorem 2.10. Suppose that a knot \( K \) can be expressed as \( K = P + Q \), where \( P \) is a prime knot, and that \( K \) can also be expressed as \( K = {K}_{1} + {K}_{2} \) . Then either\n\n(a) \( {K}_{1} = P + {K}_{1}^{\prime } \) for some \( {K}_{1}^{\prime } \), and \( Q = {K}_{1}^{\prime } + {K}_{2} \), or\n... | Proof. Let \( \sum \) be a 2-sphere in \( {S}^{3} \), meeting \( K \) transversely at two points, that demonstrates \( K \) as the sum \( {K}_{1} + {K}_{2} \) . The factorisation \( K = P + Q \) implies that there is a 3-ball \( B \) contained in \( {S}^{3} \) such that \( B \cap K \) is an arc \( \alpha \) (with \( K ... | Yes |
Corollary 2.11. Suppose that \( P \) is a prime knot and that \( P + Q = {K}_{1} + {K}_{2} \) . Suppose also that \( P = {K}_{1} \) . Then \( Q = {K}_{2} \) . | Proof. By Theorem 2.10, there are two possibilities. The first is that for some \( {K}_{1}^{\prime }, P + {K}_{1}^{\prime } = {K}_{1} = P \) and \( Q = {K}_{1}^{\prime } + {K}_{2} \) . But then the genus of \( {K}_{1}^{\prime } \) must be zero, so \( {K}_{1}^{\prime } \) is the unknot and so \( Q = {K}_{2} \) . The sec... | Yes |
Theorem 2.12. Up to ordering of summands, there is a unique expression for a knot \( K \) as a finite sum of prime knots. | Proof. Suppose \( K = {P}_{1} + {P}_{2} + \cdots + {P}_{m} = {Q}_{1} + {Q}_{2} + \cdots + {Q}_{n} \), where the \( {P}_{i} \) and \( {Q}_{i} \) are all prime. By the theorem, \( {P}_{1} \) is a summand of \( {Q}_{1} \) or of \( {Q}_{2} + \) \( {Q}_{3} + \cdots + {Q}_{n} \), and if the latter, then it is a summand of on... | Yes |
Lemma 3.2. If a diagram is changed by a Type I Reidemeister move, its bracket polynomial changes in the following way:\n\n\[ \langle {\tau }_{0} - \rangle = - {A}^{3}\langle \frown \rangle ,\;\langle - \sigma \rangle = - {A}^{-3}\langle \frown \rangle . \] | Proof.\n\n\[ \langle {\tau }^{ - }\rangle = A\langle \widehat{\sigma }\rangle + {A}^{-1}\langle \tau \rangle \]\n\n\[ = \left( {A\left( {-{A}^{-2} - {A}^{2}}\right) + {A}^{-1}}\right) \langle \frown \rangle \text{.} \]\n\nThat produces the first equation; the second follows in the same way. | Yes |
Lemma 3.3. If a diagram \( D \) is changed by a Type II or Type III Reidemeister move, then \( \langle D\rangle \) does not change. That is,\n\n(i) \( \langle \) , \( > < \rangle = \langle > < \rangle \) , (ii) \( \langle z < < \rangle = \langle z < < \rangle \) .\n\nHence \( \langle D\rangle \) is invariant under regu... | Proof. (i)\n\n\[ \langle > < > \rangle = A\langle > > < \rangle + {A}^{-1}\langle > < \rangle \]\n\n\[ = - {A}^{-2}\langle \rangle \langle \rangle + \langle > \langle \rangle + {A}^{-2}\langle \rangle \langle \rangle . \]\n\n(ii)\n\n\[ \langle x < y\rangle = A\langle x < y\rangle + {A}^{-1}\langle x > y\rangle \]\n\n\[... | No |
Theorem 3.5. Let \( D \) be a diagram of an oriented link \( L \) . Then the expression\n\n\[ \n{\left( -A\right) }^{-{3w}\left( D\right) }\langle D\rangle \n\]\n\nis an invariant of the oriented link \( L \) . | Proof. It follows from Lemma 3.3 that the given expression is unchanged by Reidemeister moves of Types II and III; Lemma 3.2 and the above remarks on \( w\left( D\right) \) show it is unchanged by a Type I move. As any two diagrams of two equivalent links are related by a sequence of such moves, the result follows at o... | Yes |
Proposition 3.7. The Jones polynomial invariant is a function\n\n\[ V : \\left\\{ {\\text{ Oriented links in }{S}^{3}}\\right\\} \\rightarrow \\mathbb{Z}\\left\\lbrack {{t}^{-1/2},{t}^{1/2}}\\right\\rbrack \]\n\n such that\n\n(i) \( V \) (unknot) \( = 1 \) ,\n\n(ii) whenever three oriented links \( {L}_{ + },{L}_{ - } ... | Proof.\n\n\[ \\langle X\\rangle = A\\langle X\\rangle + {A}^{-1}\\langle X\\rangle \]\n\n\[ \\langle X\\rangle = {A}^{-1}\\langle X\\rangle + A\\langle X\\rangle . \]\n\nMultiplying the first equation by \( A \), the second by \( {A}^{-1} \), and subtracting gives\n\n\[ A\\langle > < \\rangle - {A}^{-1}\\langle > < \\r... | Yes |
Lemma 4.5. Let \( D \) be a non-split diagram for \( L \) . Suppose that \( F \) is a 2-sphere with the property that it separates the components of \( L \) ; then \( F \) can be replaced by another 2-sphere with the same property that is in standard position. | Proof. (a) Suppose that \( C \) is amongst the \( n \) components of \( F \cap {S}_{ + } \) that do not bound disc components of \( F \cap {B}_{ + } \) . Choose \( C \) to be innermost on \( {S}_{ + } \) amongst such components. Then \( C \) is the boundary of a disc \( \Delta \) in \( {S}_{ + } \), and any component o... | Yes |
Lemma 5.4. Let \( D \) be a link diagram with \( n \) crossings. Then\n\n(i) \( M\langle D\rangle \leq n + 2\left| {{s}_{ + }D}\right| - 2 \), with equality if \( D \) is plus-adequate, and\n\n(ii) \( m\langle D\rangle \geq - n - 2\left| {{s}_{ - }D}\right| + 2 \), with equality if \( D \) is minus-adequate. | Proof. (This is due, essentially, to Kauffman.) For any state \( s \) for \( D \) let\n\n\[ \langle D \mid s\rangle = {A}^{\mathop{\sum }\limits_{{i = 1}}^{n}s\left( i\right) }{\left( -{A}^{-2} - {A}^{2}\right) }^{\left| {sD}\right| - 1}, \]\n\nso that \( \langle D\rangle = \mathop{\sum }\limits_{s}\langle D \mid s\ran... | Yes |
Lemma 5.6. Let \( D \) be a connected link diagram with \( {n}_{ \cdot } \) crossings. Then\n\n\[ \left| {{s}_{ + }D}\right| + \left| {{s}_{ - }D}\right| \leq n + 2 \] | Proof. Use induction on \( n \) . The result is clearly true when \( n = 0 \) ; suppose it to be true for diagrams with \( n - 1 \) crossings. Select a crossing of \( D \) . For at least one of the two ways of replacing the crossing with two segments that do not cross, the resulting diagram \( {D}^{\prime } \) is conne... | Yes |
Lemma 5.7. Let \( D \) be a connected \( n \) -crossing diagram.\n\n(i) If \( D \) is alternating, then \( \left| {{s}_{ + }D}\right| + \left| {{s}_{ - }D}\right| = n + 2 \). | Proof. When \( D \) is alternating, \( \left| {{s}_{ + }D}\right| + \left| {{s}_{ - }D}\right| \) is the number of planar regions in the complement of \( D \) (as \( \left| {{s}_{ + }D}\right| \) is the number of black regions, \( \left| {{s}_{ - }D}\right| \) the number of white regions in a chessboard colouring). How... | Yes |
Theorem 5.9. Let \( D \) be a connected, \( n \) -crossing diagram of an oriented link \( L \) with Jones polynomial \( V\left( L\right) \) . Then\n\n(i) \( B\left( {V\left( L\right) }\right) \leq n \) ;\n\n(ii) if \( D \) is alternating and reduced, then \( B\left( {V\left( L\right) }\right) = n \) ;\n\n(iii) if \( D ... | Proof. Recall that under the substitution \( t = {A}^{-4} \) the Jones polynomial is given by \( V\left( L\right) = {\left( -A\right) }^{-{3w}\left( D\right) }\langle D\rangle \), so that \( {4B}\left( {V\left( L\right) }\right) = B\langle D\rangle = M\langle D\rangle - m\langle D\rangle \) (where \( M\langle D\rangle ... | Yes |
Corollary 5.10. If a link \( L \) has a connected, reduced, alternating diagram of \( n \) crossings, then it has no diagram of less than n crossings; any non-alternating prime diagram for \( L \) has more than \( n \) crossings. | Proof. The existence of the reduced alternating diagram for \( L \) implies, using Theorem 5.9 (ii), that \( B\left( {V\left( L\right) }\right) = n \) . If \( L \) has another diagram of \( m \) crossings, then Theorem 5.9 (i) implies that \( n = B\left( {V\left( L\right) }\right) \leq m \) . If this second diagram is ... | Yes |
Lemma 5.12. If \( D \) is plus-adequate, then \( {D}^{r} \) is plus-adequate; if \( D \) is minus-adequate, then \( {D}^{r} \) is minus-adequate. | Proof. The result is immediate, because \( {s}_{ + }\left( {D}^{r}\right) = {\left( {s}_{ + }D\right) }^{r} \) ; see Figure 5.4. If \( D \) is plus-adequate, no component of \( {s}_{ + }\left( {D}^{r}\right) \) abuts itself at a former crossing, as it runs parallel to a component of \( {s}_{ + }D \) which, itself, has ... | Yes |
Theorem 5.13. Let \( D \) and \( E \) be diagrams, with \( {n}_{D} \) and \( {n}_{E} \) crossings respectively, for the same oriented link \( L \). Suppose that \( D \) is plus-adequate; then\n\n\[ \n{n}_{D} - w\left( D\right) \leq {n}_{E} - w\left( E\right) \n\] | Proof. Let \( \left\{ {L}_{i}\right\} \) be the components of \( L \), and let \( {D}_{i} \) and \( {E}_{i} \) be the subdiagrams of \( D \) and \( E \) corresponding to \( {L}_{i} \). Choose non-negative integers \( {\mu }_{i} \) and \( {v}_{i} \) such that for each \( i, w\left( {D}_{i}\right) + {\mu }_{i} = w\left( ... | Yes |
Corollary 5.14. Let \( D \) and \( E \) be as above.\n\n(i) The number of negative crossings of \( D \) is less than or equal to the number of negative crossings of \( E \) .\n\n(ii) The number of positive crossings in a minus-adequate diagram is minimal.\n\n(iii) An adequate diagram has the minimal number of crossings... | The corollary is just restating the theorem in different ways. An example of the use of the corollary is the two famous diagrams (the Perko pair), originally labelled \( {10}_{161} \) and \( {10}_{162} \), shown in Figure 3.1. The diagrams \( {10}_{161} \) and \( \overline{{10}_{162}} \) represent the same knot. Observ... | Yes |
Theorem 6.1. Any two presentation matrices \( A \) and \( {A}_{1} \) for \( M \) differ by a sequence of matrix moves of the following forms and their inverses:\n\n(i) Permutation of rows or columns;\n\n(ii) Replacement of the matrix \( A \) by \( \left( \begin{array}{ll} A & 0 \\ 0 & 1 \end{array}\right) \) ;\n\n(iii)... | Proof. Suppose that the matrices \( A \) and \( {A}_{1} \) correspond, with respect to some bases, to the maps \( \alpha \) and \( {\alpha }_{1} \) in the following presentations:\n\n\[ F\overset{\alpha }{ \rightarrow }E\overset{\phi }{ \rightarrow }M \rightarrow 0 \]\n\n\[ \downarrow \gamma \; \downarrow \beta \; \upd... | Yes |
Proposition 6.3. Suppose that \( F \) is a connected, compact, orientable surface with non-empty boundary, piecewise linearly contained in \( {S}^{3} \) . Then the homology groups \( {H}_{1}\left( {{S}^{3} - F;\mathbb{Z}}\right) \) and \( {H}_{1}\left( {F;\mathbb{Z}}\right) \) are isomorphic, and there is a unique nons... | Proof. The surface \( F \) is now embedded in \( {S}^{3} \) . As before, \( {H}_{1}\left( {F;\mathbb{Z}}\right) = \) \( {\bigoplus }_{{2g} + n - 1}\mathbb{Z} \) generated by \( \left\{ \left\lbrack {f}_{i}\right\rbrack \right\} \) . Let \( V \) be a regular neighbourhood of \( F \) in \( {S}^{3} \), so that \( V \) is ... | Yes |
Theorem 6.5. Let \( F \) be a Seifert surface for an oriented link \( L \) in \( {S}^{3} \) and let \( A \) be a matrix, with respect to any basis of \( {H}_{1}\left( {F;\mathbb{Z}}\right) \), for the corresponding Seifert form. Then \( {tA} - {A}^{\tau } \) is a matrix that presents the \( \mathbb{Z}\left\lbrack {{t}^... | Proof. Express \( {X}_{\infty } \) as the union of subspaces \( {Y}^{\prime } \) and \( {Y}^{\prime \prime } \), where \( {Y}^{\prime } = \mathop{\bigcup }\limits_{i}{Y}_{{2i} + 1} \) and \( {Y}^{\prime \prime } = \mathop{\bigcup }\limits_{i}{Y}_{2i} \) . Each of these subspaces is the disjoint union of countably many ... | Yes |
Theorem 6.10.\n\n(i) For any oriented link \( L,{\Delta }_{L}\left( t\right) \doteq {\Delta }_{L}\left( {t}^{-1}\right) \).\n\n(ii) For any (oriented) knot \( K,{\Delta }_{K}\left( 1\right) = \pm 1 \). | Proof. (i) Suppose that \( A \) is an \( n \times n \) Seifert matrix for \( L \) . Then\n\n\[ \n{\Delta }_{L}\left( t\right) \doteq \det \left( {{tA} - {A}^{\tau }}\right) = \det \left( {t{A}^{\tau } - A}\right) = {\left( -t\right) }^{n}\det \left( {{t}^{-1}A - {A}^{\tau }}\right) \doteq {\Delta }_{L}\left( {t}^{-1}\r... | Yes |
Corollary 6.11. For any knot \( K \) ,\n\n\[{\Delta }_{K}\left( t\right) \doteq {a}_{0} + {a}_{1}\left( {{t}^{-1} + t}\right) + {a}_{2}\left( {{t}^{-2} + {t}^{2}}\right) + \cdots ,\]\n\nwhere the \( {a}_{i} \) are integers and \( {a}_{0} \) is odd. | Proof. By Theorem 6.10(i), \( {\Delta }_{K}\left( t\right) \) can be written in the form \( {\Delta }_{K}\left( t\right) = {b}_{0} + \) \( {b}_{1}t + {b}_{2}{t}^{2} + \cdots + {b}_{N}{t}^{N} \), where \( {b}_{N - r} = \pm {b}_{r} \) with the same choice of sign for all \( r \) . If \( N \) were odd, \( {\Delta }_{K}\le... | Yes |
Proposition 6.12. Let \( L \) be an oriented link. Then \( \bar{L} \) and \( \mathrm{r}L \), the reflection and the reverse of \( L \), have the same Alexander polynomial as \( L \) up to multiplication by units. | Proof. If \( A \) is a Seifert matrix for \( L, - A \) is a Seifert matrix for \( \bar{L} \) and \( {A}^{\tau } \) is a Seifert matrix for \( \mathrm{r}L \) . | Yes |
Proposition 6.13. If a knot \( K \) has genus \( g \), then \( {2g} \geq \) breadth \( {\Delta }_{K}\left( t\right) \) . | Proof. Let \( F \) be a genus \( g \) Seifert surface for \( K \) . Then \( {tA} - {A}^{\tau } \) is a \( {2g} \times {2g} \) matrix, and so the degree in \( t \) of the polynomial \( \det \left( {{tA} - {A}^{\tau }}\right) \) is at most \( {2g} \) . | Yes |
Proposition 6.14. Suppose an oriented link \( L \) bounds a disconnected oriented surface in \( {S}^{3} \) ; then \( {\Delta }_{L}\left( t\right) \) is the zero polynomial. | Proof. Suppose \( \sum \) is a disconnected oriented surface with boundary \( L \) . Form a connected surface \( F \) by connecting the components of \( \sum \) together with thin \ | No |
In \( {S}^{3} \), let \( T \) be a standard, unknotted, solid torus that contains a knot \( K \). Let \( e : T \rightarrow {S}^{3} \) be an embedding of \( T \) onto a neighbourhood of a knot \( C \), so that e maps a longitude of \( T \) (coming from the inclusion of \( T \) in \( {S}^{3} \)) onto a longitude of \( C ... | \[ {\Delta }_{eK}\left( t\right) \doteq {\Delta }_{K}\left( t\right) {\Delta }_{C}\left( {t}^{n}\right) \] where \( K \) represents \( n \) times a generator of \( {H}_{1}\left( T\right) \). Proof. Construct Seifert surfaces for the pattern knot \( K \) and the satellite \( {eK} \) in the following way: The unknotted s... | Yes |
Theorem 6.17. Let \( K \) be a knot in \( {S}^{3} \) and let \( t : {X}_{\infty } \rightarrow {X}_{\infty } \) be the (covering) translation of \( {X}_{\infty } \) (the infinite cyclic cover of the exterior of \( K \) ). Then \( {H}_{1}\left( {{X}_{\infty };\mathbb{Q}}\right) \) is a finite-dimensional vector space ove... | Proof. The ring \( \mathbb{Q}\left\lbrack {{t}^{-1}, t}\right\rbrack \) is a principal ideal domain. A proof of this, using the Euclidean algorithm, is much the same as the proof that shows the ring of ordinary polynomials over a field to be a principal ideal domain. Over \( \mathbb{Q}\left\lbrack {{t}^{-1}, t}\right\r... | Yes |
Lemma 7.4. A covering map \( p : E \rightarrow B \) has the path lifting property. That is, given a point \( {e}_{0} \in E \) and a continuous map \( f : \left\lbrack {0,1}\right\rbrack \rightarrow B \) such that \( f\left( 0\right) = p\left( {e}_{0}\right) \), there exists a unique continuous map \( \widehat{f} : \lef... | Proof. The space \( B \) is the union of open sets \( \{ V\} \), as in the definition of a covering. Thus, by the compactness of \( \left\lbrack {0,1}\right\rbrack \) there is a dissection \( 0 = {t}_{0} < {t}_{1} < \) \( {t}_{2} < \cdots < {t}_{n} = 1 \) so that \( f\left\lbrack {{t}_{i - 1},{t}_{i}}\right\rbrack \sub... | Yes |
Lemma 7.5. A covering map \( p : E \rightarrow B \) has homotopy-lifting property for paths. That is, given a continuous map \( \widehat{f} : \left\lbrack {0,1}\right\rbrack \times \{ 0\} \rightarrow E \) and a continuous map \( f : \left\lbrack {0,1}\right\rbrack \times \left\lbrack {0,1}\right\rbrack \rightarrow B \)... | Proof. The proof of this is entirely analogous to the proof of the previous lemma; here a dissection of the square \( \left\lbrack {0,1}\right\rbrack \times \left\lbrack {0,1}\right\rbrack \) into a mesh of small squares, each mapping into some \( {V}_{i} \), is used. | No |
Proposition 7.7. Let \( p : E \rightarrow B \) be a covering map with base points \( {e}_{0} \in E \) and \( {b}_{0} \in B \), chosen so that \( p{e}_{0} = {b}_{0} \). Suppose \( X \) is a path-connected, locally path-connected, space with base point \( {x}_{0} \), and let \( f : \left( {X,{x}_{0}}\right) \rightarrow \... | Proof. If \( g \) exists, then \( {p}_{ \star }{g}_{ \star } = {f}_{ \star } \), and the result is clear. Conversely, suppose \( {f}_{ \star }{\Pi }_{1}\left( {X,{x}_{0}}\right) \subset {p}_{ \star }{\Pi }_{1}\left( {E,{e}_{0}}\right) \). If \( x \in X \), choose a path \( \alpha : \left\lbrack {0,1}\right\rbrack \righ... | Yes |
Proposition 7.8. Suppose \( p : \left( {E,{e}_{0}}\right) \rightarrow \left( {B,{b}_{0}}\right) \) and \( {p}^{\prime } : \left( {{E}^{\prime },{e}_{0}^{\prime }}\right) \rightarrow \left( {B,{b}_{0}}\right) \) are two based coverings of \( B \) with the same group. Then these are equivalent in the sense that there exi... | Proof. By Proposition 7.7, the map \( {p}^{\prime } \) lifts to a map \( h : \left( {{E}^{\prime },{e}_{0}^{\prime }}\right) \rightarrow \left( {E,{e}_{0}}\right) \) such that \( {ph} = {p}^{\prime } \) . Similarly, by Proposition 7.7 applied to the map \( p \) and covering \( {p}^{\prime } \), there is a map \( {h}^{\... | Yes |
Theorem 7.9. The covering space \( p : {X}_{\infty } \rightarrow X \) of the exterior \( X \) of an oriented link \( L \) does not depend on the choice of Seifert surface used in its construction. Further, the action of the infinite cyclic group on \( {X}_{\infty } \) is likewise independent of \( F \) . | Proof. It is clear from the construction of \( {X}_{\infty } \) that a loop \( \alpha : \left\lbrack {0,1}\right\rbrack \rightarrow X \) lifts to a loop \( \widehat{\alpha } \) (that is, \( \widehat{\alpha }\left( 0\right) = \widehat{\alpha }\left( 1\right) \) ) in \( {X}_{\infty } \) provided \( \widehat{\alpha }\left... | Yes |
Theorem 7.10. If the rth elementary ideal of the Alexander module of a knot \( K \) is not the whole of \( \mathbb{Z}\left\lbrack {{t}^{-1}, t}\right\rbrack \), then \( K \) has unknotting number \( u\left( K\right) \geq r \) . | As an example, consider the pretzel knot \( P\left( {3,3, - 3}\right) \) discussed in Example 6.9. There it was shown that the second elementary ideal of the Alexander module is not \( \mathbb{Z}\left\lbrack {{t}^{-1}, t}\right\rbrack \), and so \( u\left( {P\left( {3,3, - 3}\right) }\right) \geq 2 \) . It is easy to s... | Yes |
Theorem 7.13. Let \( B \) be a path-connected, locally path-connected, semi-locally simply connected space. Then there exists a simply connected space \( \widetilde{B} \) and covering map \( p : \widetilde{B} \rightarrow B \). Furthermore, the group \( {\Pi }_{1}\left( B\right) \) acts freely as a group of homeomorphis... | Proof. Let \( {b}_{0} \in B \) be a base point and let \( X \) be the set of all paths \( \alpha : \left\lbrack {0,1}\right\rbrack \rightarrow \) \( B \) such that \( \alpha \left( 0\right) = {b}_{0} \). Define an equivalence relation on \( \mathrm{X} \) by letting \( \alpha \sim \beta \) if and only if \( \alpha \left... | No |
Theorem 7.14. Suppose that a group \( G \) acts as a group of homeomorphisms on a path-connected, locally path-connected, space \( Y \) . Suppose that each \( y \) belonging to \( Y \) has an open neighbourhood \( U \) such that \( U \cap {gU} = \varnothing \) for all \( g \in G - \{ 1\} \) . Then the quotient map \( q... | Proof. If \( y \in Y \), there is an open neighbourhood \( U \) of \( y \) such that \( U \cap {gU} = \varnothing \) for all \( g \in G - \{ 1\} \) . Now \( {q}^{-1}\left( {qU}\right) = \mathop{\bigcup }\limits_{{g \in G}}{gU} \) . This is open because each \( {gU} \) is open (because \( g \) is a homeomorphism). Hence... | Yes |
Theorem 8.4. Let \( A \) and \( B \) be Seifert matrices for an oriented link \( L \). Then \( A \) and B are S-equivalent. | Proof. Suppose that \( A \) is an \( n \times n \) matrix corresponding to a Seifert surface \( F \), with respect to some base of \( {H}_{1}\left( {F;\mathbb{Z}}\right) \). Changing the base used for \( {H}_{1}\left( {F;\mathbb{Z}}\right) \) changes \( A \) to a matrix of the form \( {P}^{\tau }{AP} \), where \( P \) ... | Yes |
Theorem 8.5. The Conway-normalised Alexander polynomial is a well-defined invariant of the oriented link \( L \) . | Proof. It is only necessary to check the invariance of the Conway-normalised polynomial when \( A \) changes by \( S \) -equivalence. Firstly, note that\n\n\[ \det \left( {{t}^{1/2}{P}^{\tau }{AP} - {t}^{-1/2}{P}^{\tau }{A}^{\tau }P}\right) = {\left( \det P\right) }^{2}\det \left( {{t}^{1/2}A - {t}^{-1/2}{A}^{\tau }}\r... | Yes |
Theorem 8.6. For oriented links \( L \), the Conway-normalised Alexander polynomial \( {\Delta }_{L}\left( t\right) \in \mathbb{Z}\left\lbrack {{t}^{-\frac{1}{2}},{t}^{\frac{1}{2}}}\right\rbrack \) is characterised by\n\n(i) \( {\Delta }_{\text{unknot }}\left( t\right) = 1 \) ,\n\n(ii) whenever three oriented links \( ... | Proof. Construct a Seifert surface \( {F}_{0} \) for \( {L}_{0} \) that meets the neighbourhood of the point in question as shown in Figure 8.1. The Seifert circuit method described in Chapter 2 will do this. Now form Seifert surfaces \( {F}_{ + } \) for \( {L}_{ + } \) and \( {F}_{ - } \) for \( {L}_{ - } \) by adding... | Yes |
Proposition 8.7. For an oriented link \( L \) with \( \# L \) components, the Conway polynomial has the following properties.\n\n(i) If \( L \) is a split link, then \( {\nabla }_{L}\left( z\right) = 0 \) . | Proof. (i) This follows from the stronger Proposition 6.14. However, it also follows at once by applying the skein formula to links \( {L}_{ + },{L}_{ - } \) and \( {L}_{0} \) shown in Figure 8.2. As \( {L}_{ + } \) and \( {L}_{ - } \) are here the same link, \( {\nabla }_{{L}_{0}}\left( z\right) = 0 \) . | Yes |
Theorem 8.9. The \( \omega \) -signature \( {\sigma }_{\omega }\left( L\right) \) is well defined as an invariant of \( L \) . | Proof. The signature of a Hermitian matrix is not changed by congruence (that fact is Sylvester's famous law of inertia), so it is only necessary to see whether the definition changes under an elementary enlargement of a Seifert matrix \( A \) .\n\nSuppose\n\n\[ B = \left( \begin{array}{lll} A & \xi & 0 \\ 0 & 0 & 1 \\... | Yes |
Theorem 8.10. If \( L \) is an oriented link in \( {S}^{3} \) and \( \bar{L} \) is its reflection, then for any unit complex number \( \omega \neq 1 \) , \[ {\sigma }_{\omega }\left( L\right) = - {\sigma }_{\omega }\left( \bar{L}\right) \] | Proof. If \( A \) is a Seifert matrix for \( L \), then \( - A \) is a Seifert matrix for \( \bar{L} \) | No |
Lemma 8.12. Suppose that for some knot \( K \) in \( {S}^{3} \), there is a flat surface \( F \) in \( {B}^{4} \) with \( F \cap {S}^{3} = \partial F \cap {S}^{3} = K \) . Then the inclusion map induces an isomorphism \( {H}_{1}\left( {{S}^{3} - K}\right) \rightarrow {H}_{1}\left( {{B}^{4} - F}\right) \cong \mathbb{Z}.... | Proof. Let \( N \), a copy of \( F \times {I}^{2} \), be a neighbourhood of \( F \) meeting \( {S}^{3} \) in \( \partial F \times {I}^{2} \) . The Mayer-Vietoris theorem gives an exact sequence\n\n\[ 0 = {H}_{2}\left( {B}^{4}\right) \rightarrow {H}_{1}\left( {F \times \partial {I}^{2}}\right) \rightarrow {H}_{1}\left( ... | Yes |
Lemma 8.13. Suppose that \( {f}_{1} : {F}_{1} \rightarrow {B}^{4} \) and \( {f}_{2} : {F}_{2} \rightarrow {B}^{4} \) are maps, of orientable surfaces into the 4-ball, which have disjoint images. Suppose that on \( \partial {F}_{i} \) the map \( {f}_{i} \) is a homeomorphism onto a knot \( {K}_{i} \) in \( {S}^{3} = \pa... | Proof. After moving the maps into general position, it may be assumed that each \( {f}_{i} \) has only double points as singularities. That means that near the image of such a singularity in \( {B}^{4} \), the image of \( {F}_{i} \) looks like two standard planes in \( {\mathbb{R}}^{4} \) meeting in a point \( P \) . T... | Yes |
Corollary 8.16. There is a base \( \left\lbrack {f}_{1}\right\rbrack ,\left\lbrack {f}_{2}\right\rbrack ,\ldots ,\left\lbrack {f}_{2g}\right\rbrack \) over \( \mathbb{Z} \) for \( {H}_{1}\left( {\partial M;\mathbb{Z}}\right) \) so that \( \left\lbrack {f}_{1}\right\rbrack ,\left\lbrack {f}_{2}\right\rbrack ,\ldots ,\le... | Proof. One may consider \( {H}_{1}\left( {\partial M;\mathbb{Z}}\right) \) to be \( {\mathbb{Z}}^{2g} \subset {\mathbb{Q}}^{2g} = {H}_{1}\left( {\partial M;\mathbb{Q}}\right) \) . The \( g \) -dimensional subspace \( U \) of \( {\mathbb{Q}}^{2g} \), given by Lemma 8.15, has a base consisting of elements in \( {\mathbb{... | Yes |
Proposition 8.17. Suppose that \( F \) is a genus \( g \) Seifert surface for a slice knot \( K \) in \( {S}^{3} \) . Then a base may be chosen for \( {H}_{1}\left( {F;\mathbb{Z}}\right) \) with respect to which the corresponding Seifert matrix has the form\n\n\[ \left( \begin{matrix} 0 & P \\ Q & R \end{matrix}\right)... | Proof. Let \( D \) be a slicing disc for \( K \) contained in \( {B}^{4} \) . By Lemma 8.14 there is contained in \( {B}^{4} \) a 3-manifold \( M \) having an \( M \times \left\lbrack {-1,1}\right\rbrack \) neighbourhood such that \( \partial M = D \cup F \) . Corollary 8.16 gives a certain base \( \left\lbrack {f}_{1}... | Yes |
Theorem 8.18. If \( K \) is a slice knot, then the Conway-normalised Alexander polynomial of \( K \) is of the form \( f\left( t\right) f\left( {t}^{-1}\right) \), where \( f \) is a polynomial with integer coefficients. | Proof. Using the Seifert matrix of Proposition 8.17, the required Alexander polynomial is the determinant of\n\n\[ \left( \begin{matrix} 0 & {t}^{1/2}P - {t}^{-1/2}{Q}^{\tau } \\ {t}^{1/2}Q - {t}^{-1/2}{P}^{\tau } & {t}^{1/2}R - {t}^{-1/2}{R}^{\tau } \end{matrix}\right) ,\]\n\nwhich is \( \det \left( {{tP} - {Q}^{\tau ... | Yes |
Theorem 8.19. If \( K \) is a slice knot, then the signature of \( K \) is zero and, if the unit complex number \( \omega \) is not a zero of the Alexander polynomial, then \( {\sigma }_{\omega }\left( K\right) = 0 \) . | Proof. This follows at once from the fact that the signature is zero for a quadratic form coming from a non-singular symmetric bilinear form that vanishes on a subspace of half the dimension of the space concerned. A similar result holds for Hermitian forms. | Yes |
Theorem 9.1. Let \( {X}_{2} \) be the cyclic double cover of \( {S}^{3} \) branched over a link \( L \) and suppose that \( A \) is a Seifert matrix for \( L \) with respect to some orientation and some Seifert surface. Then \( {H}_{1}\left( {X}_{2}\right) \) is presented, as an abelian group, by the matrix \( \left( {... | Proof. In the above notation, \( {\widehat{X}}_{2} = {Y}_{0} \cup {Y}_{1} \), where \( {Y}_{0} \cap {Y}_{1} \) is two disjoint copies of \( F \) . A presentation of \( {H}_{1}\left( {\widehat{X}}_{2}\right) \) can be obtained from the following exact Mayer-Vietoris sequence:\n\n\[ \rightarrow {H}_{1}\left( {{Y}_{0} \ca... | Yes |
Let \( {X}_{2} \) be the double cover of \( {S}^{3} \) branched over a link \( L \) . The order of the group \( {H}_{1}\left( {X}_{2}\right) \) is the modulus of the determinant of \( \left( {A + {A}^{\tau }}\right) \), that is\n\n\[ \left| {{H}_{1}\left( {X}_{2}\right) }\right| = \left| {\det \left( {A + {A}^{\tau }}\... | Proof. Any finitely generated abelian group can be expressed as a direct sum of cyclic groups. Thus it has as a presentation matrix a diagonal matrix, the entries on the diagonal being the orders of the summands, with the convention that an infinite group has order zero. By Theorem 6.1, the determinant of a square pres... | No |
Any Goeritz matrix for a link \( L \), associated with the white regions of a diagram of \( L \), represents, with respect to some base, the Gordon-Litherland form \[ {\mathcal{G}}_{F} : {H}_{1}\left( F\right) \times {H}_{1}\left( F\right) \rightarrow \mathbb{Z}, \] where \( F \) is the spanning surface for \( L \) giv... | Let the white regions, \( {R}_{0},{R}_{1},\ldots ,{R}_{n} \), of the diagram inherit an orientation from the sphere \( {S}^{2} \) in which they are assumed to lie; thus each \( \partial {R}_{i} \) has an orientation. Let \( {f}_{i} \) be the oriented simple closed curve in \( F \) that consists of \( \partial {R}_{i} \... | Yes |
Corollary 9.5. The determinant of \( L,\left| {{\Delta }_{L}\left( {-1}\right) }\right| \), is equal to \( \left| {\det G}\right| \), where \( G \) is any Goertiz matrix for \( L \) . | The proof of this is immediate from the last three theorems. It follows that \( \left| {\det G}\right| \) is an invariant of \( L \), and, as a Goeritz matrix is often easy to write down, it can be a useful invariant. | No |
Theorem 9.6. Suppose that \( {L}_{ + },{L}_{ - },{L}_{0} \) and \( {L}_{\infty } \) are four links that have identical diagrams except near a point where they are as shown in Figure 9.5. Then\n\n\[ \n{\left( \det {L}_{ + }\right) }^{2} + {\left( \det {L}_{ - }\right) }^{2} = 2\left( {{\left( \det {L}_{0}\right) }^{2} +... | Proof. The diagram shows the four links together with connected shaded spanning surfaces \( {F}_{i} \) for \( i = + , - ,0,\infty \) . These can always be constructed by using Seifert’s method (see Chapter 2) for \( {F}_{0} \) and adding bands to get the other three surfaces. The four surfaces are taken to be identical... | Yes |
Theorem 9.7. Let \( {X}_{r} \) be the cyclic \( r \) -fold cover of \( {S}^{3} \) branched over an \( n \) - component oriented link \( L \), and suppose that \( A \) is a Seifert matrix for \( L \) coming from a genus \( g \) Seifert surface. Then \( {H}_{1}\left( {X}_{r}\right) \) is presented, as an abelian group, b... | Assuming that a \ | No |
The order of the first homology group of \( {X}_{r} \), the cyclic \( r \) -fold cover of \( {S}^{3} \) branched over \( L \), is given by\n\n\[ \left| {{H}_{1}\left( {X}_{r}\right) }\right| = \left| {\mathop{\prod }\limits_{{v = 1}}^{{r - 1}}{\Delta }_{L}\left( {e}^{{2\pi }\imath \frac{v}{r}}\right) }\right| . \] | Assuming that a \ | No |
Lemma 10.5. Suppose that \( L \) and \( {L}^{\prime } \) are oriented links having property \( \left( \star \right) \) which are the same except near one point, where they are as shown in Figure 10.1; then \( \mathcal{A}\left( L\right) = \mathcal{A}\left( {L}^{\prime }\right) \) | Proof. The two segments shown on one of the two sides of Figure 10.1 must belong to the same component of the link. Suppose, without loss of generality, it is the two segments on the left side. Then using the Seifert circuit method of Theorem 2.2, a Seifert surface can be constructed for the left link that meets the ne... | Yes |
Theorem 10.7. Let \( K \) be a knot. Then \( \mathcal{A}\left( K\right) \equiv {a}_{2}\left( K\right) \) modulo 2, where \( {a}_{2}\left( K\right) \) is the coefficient of \( {z}^{2} \) in the Conway polynomial \( {\nabla }_{K}\left( z\right) \) . The Arfinvariant of \( K \) is related to the Alexander polynomial by\n\... | Proof. The formula \( \mathcal{A}\left( {L}_{ + }\right) - \mathcal{A}\left( {L}_{ - }\right) \equiv \operatorname{lk}\left( {L}_{0}\right) \) modulo 2, valid when \( {L}_{ + } \) has one component, allows calculation of \( \mathcal{A}\left( K\right) \) from \( \mathcal{A} \) (unknot) \( = 0 \) . However, this gives th... | Yes |
Theorem 11.2. Let \( X \) be the exterior of a knot \( K \) in \( {S}^{3} \) . If \( K \) is not the unknot, then the inclusion map induces an injection \( {\Pi }_{1}\left( {\partial X}\right) \rightarrow {\Pi }_{1}\left( X\right) \) . | Proof. Suppose \( {\Pi }_{1}\left( {\partial X}\right) \rightarrow {\Pi }_{1}\left( X\right) \) is not injective. Then, by the loop theorem, there is an embedding \( e : {D}^{2} \rightarrow X \) sending \( \partial {D}^{2} \) into the torus \( \partial X \), to a simple closed curve not homotopically trivial in the tor... | Yes |
Corollary 11.3. A knot \( K \) is the unknot if and only if \( {\Pi }_{1}\left( {{S}^{3} - K}\right) \) is infinite cyclic. | Proof. If \( {\Pi }_{1}\left( {{S}^{3} - K}\right) \) is isomorphic to \( \mathbb{Z} \), there can be no injection \( {\Pi }_{1}\left( {\partial X}\right) \rightarrow \) \( {\Pi }_{1}\left( X\right) \) (as \( {\Pi }_{1}\left( {\partial X}\right) \) is isomorphic to \( \mathbb{Z} \oplus \mathbb{Z} \) ). | No |
Corollary 11.4. Let \( {X}_{1} \) and \( {X}_{2} \) be the exteriors of two non-trivial knots and let M be a 3-manifold formed by identifying their boundaries together using any homeomorphism. Then the inclusion into \( M \) of the torus \( T \) that comes from the identified boundaries induces an injection \( {\Pi }_{... | Proof. This follows at once from the above theorem and from the Van Kam-pen theorem, which describes how fundamental groups behave when a space is described as a union of subspaces. | No |
Theorem 11.6. If \( K \) is a knot in \( {S}^{3} \) any map \( {S}^{2} \rightarrow {S}^{3} - K \) is homotopic to a constant map (that is, \( {\Pi }_{2}\left( {{S}^{3} - K}\right) = 0 \) ). | Proof. If the statement is false then, by the sphere theorem, there exists a piecewise linear embedding \( e : {S}^{2} \rightarrow {S}^{3} - K \) that is not homotopic to a constant in \( \left( {{S}^{3} - K}\right) \) . Then, by the Schönflies theorem, \( e\left( {S}^{2}\right) \) separates \( {S}^{3} \) into two comp... | Yes |
Theorem 11.7. If \( K \) is a knot in \( {S}^{3} \), any map \( {S}^{r} \rightarrow {S}^{3} - K \) is homotopic to a constant map (that is, \( {\Pi }_{r}\left( {{S}^{3} - K}\right) = 0 \) ) for all \( r \geq 2 \) . | Proof. Let \( X \) be the exterior of \( K \) and let \( \widetilde{X} \) be the universal cover of \( X \) . Thus \( \widetilde{X} \) is the simply connected cover of \( X \), it is acted upon by \( {\Pi }_{1}\left( X\right) \), and the quotient of \( \widetilde{X} \) by this action is \( X \) . The operation of lifti... | Yes |
Theorem 11.9. If \( {K}_{1} \) and \( {K}_{2} \) are prime knots in \( {S}^{3} \) and \( {\Pi }_{1}\left( {{S}^{3} - {K}_{1}}\right) \) and \( {\Pi }_{1}\left( {{S}^{3} - }\right. \) \( \left. {K}_{2}\right) \) are isomorphic groups, then \( \left( {{S}^{3} - {K}_{1}}\right) \) and \( \left( {{S}^{3} - {K}_{2}}\right) ... | Thus, for prime knots, the knot group determines the complement of the knot. It is by no means obvious that this means that the knots are the same. Perhaps the homeomorphism might send a meridian to a non-meridian. That this is not so is the substance of one of the most impressive results in knot theory of the 1980's. ... | No |
Lemma 12.2. Suppose that \( U \) and \( V \) are 3-manifolds with homeomorphic boundaries, and that \( {h}_{0} : \partial U \rightarrow \partial V \) and \( {h}_{1} : \partial U \rightarrow \partial V \) are isotopic homeomorphisms. Then \( U{ \cup }_{{h}_{0}}V \) and \( U{ \cup }_{{h}_{1}}V \) are homeomorphic. | Proof. Choose ([113],[47]) a collar neighbourhood \( C \) of \( \partial U \) in \( U \) ; \( C \) is a neighbourhood of \( \partial U \) homeomorphic to \( \partial U \times \left\lbrack {0,1}\right\rbrack \), with \( \partial U \) identified with \( \partial U \times 0 \) . A homeomorphism \( f : U{ \cup }_{{h}_{0}}V... | Yes |
Lemma 12.5. Suppose oriented simple closed curves \( p \) and \( q \), contained in the interior of the surface \( F \), intersect transversely at precisely one point. Then \( p{ \sim }_{\tau }q \) . | Proof. The first diagram of Figures 12.3 shows the intersection point of \( p \) and \( q \) and also a simple closed curve \( {C}_{1} \) that runs parallel to, and is slightly displaced from, \( q \) . Similarly, \( {C}_{2} \) is a slightly displaced copy of \( p \) . The second diagram shows \( {\tau }_{1}p \), where... | Yes |
Lemma 12.6. Suppose that oriented simple closed curves \( p \) and \( q \) contained in the interior of the surface \( F \) are disjoint and that neither separates \( F \) (that is, \( \left\lbrack p\right\rbrack \neq 0 \neq \left\lbrack q\right\rbrack \) in \( \left. {{H}_{1}\left( {F,\partial F}\right) }\right) \) . ... | Proof. Consideration of the surface obtained by cutting \( F \) along \( p \cup q \) shows at once that there is a simple closed curve \( r \) in \( F \) that intersects each of \( p \) and \( q \) transversely at one point. Then, by Lemma 12.5, \( p{ \sim }_{\tau }r{ \sim }_{\tau }q \) . | Yes |
Proposition 12.7. Suppose that oriented simple closed curves \( p \) and \( q \) are contained in the interior of the surface \( F \) and that neither separates \( F \) . Then \( p{ \sim }_{\tau }q \) . | Proof. Changing \( q \) by means of a homeomorphism of \( F \) that is (close to and) isotopic to the identity, it can be assumed that \( p \) and \( q \) intersect transversely at \( n \) points. The proof is by induction on \( n \) ; Lemmas 12.5 and 12.6 start the induction, so assume that \( n \geq 2 \) and that the... | Yes |
Corollary 12.8. Let \( {p}_{1},{p}_{2},\ldots ,{p}_{n} \) be disjoint simple closed curves in the interior of \( F \) the union of which does not separate \( F \) . Let \( {q}_{1},{q}_{2},\ldots ,{q}_{n} \) be another set of curves with the same properties. Then there is a homeomorphism \( h \) of \( F \) that is in th... | Proof. Suppose inductively that such an \( h \) can be found so that \( h{p}_{i} = {q}_{i} \) for each \( i = 1,2,\ldots, n - 1 \) . Apply Proposition 12.7 to \( h{p}_{n} \) and \( {q}_{n} \) in \( F \) cut along \( {q}_{1} \cup {q}_{2} \cup \ldots \cup {q}_{n - 1} \) | Yes |
Lemma 12.12. Any closed connected orientable 3-manifold has a Heegaard splitting. | Proof. This is similar to the first part of the proof of Theorem 8.2. Take a triangulation of \( M \) as a simplicial complex \( K \) . The vertices of the first derived subdivision \( {K}^{\left( 1\right) } \) of \( K \) are the barycentres \( \widehat{A} \) of the simplexes \( A \) of \( K \) . The second derived sub... | Yes |
Theorem 12.13. Let \( M \) be a closed connected orientable 3-manifold. There exists finite sets of disjoint solid tori \( {T}_{1}^{\prime },{T}_{2}^{\prime },\ldots ,{T}_{N}^{\prime } \) in \( M \) and \( {T}_{1},{T}_{2},\ldots ,{T}_{N} \) in \( {S}^{3} \) such that \( M - { \cup }_{1}^{N}\operatorname{Int}\left( {T}_... | Proof. By Lemma 12.12, \( M \) has a Heegaard splitting, so for handlebodies \( U \) and \( V \) of some genus \( g \), and some homeomorphism \( h : \partial U \rightarrow \partial V, M = \) \( U{ \cup }_{h}V \) . Let \( {p}_{1}^{\prime },{p}_{2}^{\prime },\ldots ,{p}_{g}^{\prime } \) be disjoint simple closed curves ... | No |
Theorem 12.15. Two framed links in \( {S}^{3} \) give, by surgery, the same oriented 3- manifold if and only if they are related by a sequence of moves of two types. In a move of type 1, an extra unknotted component, unlinked from all other components, with framing 1 or -1 is added to or removed from the link. In a mov... | For the proof, which uses 4-dimensional Cerf theory, refer to [65]. If one considers the surgery information as a recipe for adding 2-handles on to a 4-ball to create a 4-manifold with the 3-manifold as its boundary, a move of type 2 corresponds to sliding one 2-handle over another. A type 1 move changes the 4-manifold... | No |
Lemma 13.2. Suppose that \( {A}^{4} \) is not a \( {k}^{\text{th }} \) root of unity for \( k \leq n \) . Then there is a unique element \( {f}^{\left( n\right) } \in T{L}_{n} \) such that\n\n(i) \( {f}^{\left( n\right) }{e}_{i} = 0 = {e}_{i}{f}^{\left( n\right) } \) for \( 1 \leq i \leq n - 1 \) ,\n\n(ii) \( \left( {{... | Proof. Note that if \( {f}^{\left( n\right) } \) exists, \( \mathbf{1} - {f}^{\left( n\right) } \) is the identity of the algebra generated by \( \left\{ {{e}_{1},{e}_{2},\ldots ,{e}_{n - 1}}\right\} \), and so \( {f}^{\left( n\right) } \) is then certainly unique. Let \( {f}^{\left( 0\right) } \) be the empty diagram ... | Yes |
Lemma 13.4. In \( \mathcal{S}\left( {{S}^{1} \times I,2\text{points}}\right) ,{a\omega } - {b\omega } \) is a linear sum of two elements, each of which contains a copy of \( {f}^{\left( r - 1\right) } \) . (That is, each of the two elements is the image of \( {f}^{\left( r - 1\right) } \) under some map \( T{L}_{r - 1}... | Proof. Consider the inclusion, shown in Figure 13.10, of the \( T{L}_{n + 1} \) recurrence relation of Figure 13.6 into the annulus.\n\n\n\nFigure 13.10\n\nThe top boundary points on either side of the square are joi... | Yes |
Lemma 13.5. Suppose that \( A \) is chosen so that \( {A}^{4} \) is a primitive \( {r}^{\text{th }} \) root of unity, \( r \geq 3 \) . Suppose that \( D \) is a planar diagram of a link of \( n \) (ordered) components. Suppose that \( {D}^{\prime } \) is another such diagram, obtained from \( D \) by a Kirby type 2 mov... | Proof. It must be checked that the elements of \( \mathcal{S}\left( {\mathbb{R}}^{2}\right) \), produced as described above from \( D \) and from \( {D}^{\prime } \), with \( \omega \) as the \ | No |
Theorem 13.8. Suppose that a closed oriented 3-manifold \( M \) is obtained by surgery on a framed link that is represented by a planar diagram D. Let \( {b}_{ + } \) be the number of positive eigenvalues and \( {b}_{ - } \) be the number negative eigenvalues of the linking matrix of this link. Suppose \( r \geq 3 \) a... | Proof. Note that \( A \) is a primitive \( 4{r}^{th} \) root of unity, and so, by Lemma 13.7, \( < \omega { > }_{{U}_{ + }} \) and \( < \omega { > }_{{U}_{ - }} \) are non-zero. It follows from the Corollary 13.6 and the preceding remarks about the linking matrix that the given expression is invariant under Kirby type ... | Yes |
Theorem 13.8. Suppose that a closed oriented 3-manifold \( M \) is obtained by surgery on a framed link that is represented by a planar diagram D. Let \( {b}_{ + } \) be the number of positive eigenvalues and \( {b}_{ - } \) be the number negative eigenvalues of the linking matrix of this link. Suppose \( r \geq 3 \) a... | Proof. Note that \( A \) is a primitive \( 4{r}^{th} \) root of unity, and so, by Lemma 13.7, \( < \omega { > }_{{U}_{ + }} \) and \( < \omega { > }_{{U}_{ - }} \) are non-zero. It follows from the Corollary 13.6 and the preceding remarks about the linking matrix that the given expression is invariant under Kirby type ... | Yes |
Lemma 13.9. Suppose \( r \geq 3 \) and \( A \) is a primitive \( 4{r}^{\text{th }} \) root of unity. The element of \( T{L}_{n} \) shown in Figure 13.13 is the zero map of outsides if \( 1 \leq n \leq r - 2 \) . When \( n = 0 \), the element acts as multiplication by \( \langle \omega {\rangle }_{U} \) . | Proof. Consider first the element of \( T{L}_{n} \) that consists of \( {f}^{\left( n\right) } \) encircled by one simple closed curve. This is shown in Figure 13.14 for \( n = 4 \) . Figure 13.14 shows a calculation for that element. Firstly one crossing is removed in the two standard ways, the results being multiplie... | Yes |
Lemma 14.2. The element of \( T{L}_{n} \) shown in Figure 14.2, which consists of the idempotent \( {f}^{\left( n\right) } \) with all its strands encircled by a parallel strands that join up the ends of an idempotent \( {f}^{\left( a\right) } \), is\n\n\[ \n{\left( -1\right) }^{a}\frac{{A}^{2\left( {n + 1}\right) \lef... | Proof. The \( a \) parallel strands and the idempotent \( {f}^{\left( a\right) } \) can, as explained in Chapter 13, be thought of as \( {S}_{a}\left( \alpha \right) \) contained in an annulus encircling the strands of \( {f}^{\left( n\right) } \), where \( {S}_{a} \) is the \( {a}^{\text{th }} \) Chebyshev polynomial.... | Yes |
Lemma 14.3. Suppose \( A \) is a primitive \( 4{r}^{\text{th }} \) root of unity. Then\n\n\[< \omega { > }_{{U}_{ + }} = \frac{G}{2{A}^{\left( 3 + {r}^{2}\right) }\left( {{A}^{2} - {A}^{-2}}\right) },\] | Proof. Recall that \( {U}_{ + } \) is the diagram of the unknot with one positive crossing,\n\n\[ \omega = \mathop{\sum }\limits_{{n = 0}}^{{r - 2}}{\Delta }_{n}{S}_{n}\left( \alpha \right) \text{ and }{\Delta }_{n} = \frac{{\left( -1\right) }^{n}\left( {{A}^{2\left( {n + 1}\right) } - {A}^{-2\left( {n + 1}\right) }}\r... | Yes |
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