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Proposition 11.3.1. For every constant \( p \in \left( {0,1}\right) \) and every graph \( H \) , almost every \( G \in \mathcal{G}\left( {n, p}\right) \) contains an induced copy of \( H \) . | Proof. Let \( H \) be given, and \( k \mathrel{\text{:=}} \left| H\right| \) . If \( n \geq k \) and \( U \subseteq \{ 0,\ldots, n - 1\} \) is a fixed set of \( k \) vertices of \( G \), then \( G\left\lbrack U\right\rbrack \) is isomorphic to \( H \) with a certain probability \( r > 0 \) . This probability \( r \) de... | Yes |
Lemma 11.3.2. For every constant \( p \in \left( {0,1}\right) \) and \( i, j \in \mathbb{N} \), almost every graph \( G \in \mathcal{G}\left( {n, p}\right) \) has the property \( {\mathcal{P}}_{i, j} \) . | Proof. For fixed \( U, W \) and \( v \in G - \left( {U \cup W}\right) \), the probability that \( v \) is adjacent to all the vertices in \( U \) but to none in \( W \), is\n\n\[ \n{p}^{\left| U\right| }{q}^{\left| W\right| } \geq {p}^{i}{q}^{j} \n\]\n\nHence, the probability that no suitable \( v \) exists for these \... | Yes |
Corollary 11.3.3. For every constant \( p \in \left( {0,1}\right) \) and \( k \in \mathbb{N} \), almost every graph in \( \mathcal{G}\left( {n, p}\right) \) is \( k \) -connected. | Proof. By Lemma 11.3.2, it is enough to show that every graph in \( {\mathcal{P}}_{2, k - 1} \) is \( k \) -connected. But this is easy: any graph in \( {\mathcal{P}}_{2, k - 1} \) has order at least \( k + 2 \), and if \( W \) is a set of fewer than \( k \) vertices, then by definition of \( {\mathcal{P}}_{2, k - 1} \... | Yes |
Proposition 11.3.4. For every constant \( p \in \left( {0,1}\right) \) and every \( \epsilon > 0 \) , almost every graph \( G \in \mathcal{G}\left( {n, p}\right) \) has chromatic number\n\n\[ \chi \left( G\right) > \frac{\log \left( {1/q}\right) }{2 + \epsilon } \cdot \frac{n}{\log n}. \] | Proof. For any fixed \( n \geq k \geq 2 \), Lemma 11.1.2 implies\n\n\[ P\left\lbrack {\alpha \geq k}\right\rbrack \leq \left( \begin{array}{l} n \\ k \end{array}\right) {q}^{\left( \begin{array}{l} k \\ 2 \end{array}\right) }\n\n\[ \leq {n}^{k}{q}^{\left( \begin{matrix} k \\ 2 \end{matrix}\right) }\n\n\[ = {q}^{k\frac{... | Yes |
Theorem 11.3.5. (Erdős and Rényi 1963)\n\nWith probability 1, a random graph \( G \in \mathcal{G}\left( {{\aleph }_{0}, p}\right) \) with \( 0 < p < 1 \) is isomorphic to the Rado graph \( R \) . | Proof. Given fixed disjoint finite sets \( U, W \subseteq \mathbb{N} \), the probability that a vertex \( v \notin U \cup W \) is not joined to \( U \cup W \) as expressed in property \( \left( *\right) \) of Chapter 8.3 (i.e., is not joined to all of \( U \) or is joined to some vertex in \( W \) ) is some number \( r... | Yes |
Lemma 11.4.1. (Chebyshev's Inequality)\n\nFor all real \( \lambda > 0 \) ,\n\n\[ P\left\lbrack {\left| {X - \mu }\right| \geq \lambda }\right\rbrack \leq {\sigma }^{2}/{\lambda }^{2} \]\n\n\( \left( {11.1.4}\right) \) | Proof. By Lemma 11.1.4 and definition of \( {\sigma }^{2} \) ,\n\n\[ P\left\lbrack {\left| {X - \mu }\right| \geq \lambda }\right\rbrack = P\left\lbrack {{\left( X - \mu \right) }^{2} \geq {\lambda }^{2}}\right\rbrack \leq {\sigma }^{2}/{\lambda }^{2}. \] | Yes |
Lemma 11.4.2. If \( \mu > 0 \) for \( n \) large, and \( {\sigma }^{2}/{\mu }^{2} \rightarrow 0 \) as \( n \rightarrow \infty \), then \( X\left( G\right) > 0 \) for almost all \( G \in \mathcal{G}\left( {n, p}\right) \) . | Proof. Any graph \( G \) with \( X\left( G\right) = 0 \) satisfies \( \left| {X\left( G\right) - \mu }\right| = \mu \) . Hence Lemma 11.4.1 implies with \( \lambda \mathrel{\text{:=}} \mu \) that\n\n\[ P\left\lbrack {X = 0}\right\rbrack \leq P\left\lbrack {\left| {X - \mu }\right| \geq \mu }\right\rbrack \leq {\sigma }... | Yes |
Corollary 11.4.4. If \( k \geq 3 \), then \( t\left( n\right) = {n}^{-1} \) is a threshold function for the property of containing a \( k \) -cycle. | Null | No |
Corollary 11.4.6. If \( k \geq 2 \), then \( t\left( n\right) = {n}^{-2/\left( {k - 1}\right) } \) is a threshold function for the property of containing a \( {K}^{k} \) . | Proof. \( {K}^{k} \) is balanced, because \( \varepsilon \left( {K}^{i}\right) = \frac{1}{2}\left( {i - 1}\right) < \frac{1}{2}\left( {k - 1}\right) = \varepsilon \left( {K}^{k}\right) \) for \( i < k \) . With \( \ell \mathrel{\text{:=}} \begin{Vmatrix}{K}^{k}\end{Vmatrix} = \frac{1}{2}k\left( {k - 1}\right) \), we ob... | Yes |
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 |
Corollary 12.1.2. If \( X \) is well-quasi-ordered, then every infinite sequence in \( X \) has an infinite increasing subsequence. | Null | No |
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 12.3.2. For every \( H \subseteq G \), the pair \( \left( {T,{\left( {V}_{t} \cap V\left( H\right) \right) }_{t \in T}}\right) \) is a tree-decomposition of \( H \) . | Null | No |
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 12.3.5. Any complete subgraph of \( G \) is contained in some part of \( \left( {T,\mathcal{V}}\right) \) . | Null | No |
Proposition 12.3.6. If \( H \preccurlyeq G \) then \( \operatorname{tw}\left( H\right) \leq \operatorname{tw}\left( G\right) \) . | Null | No |
For every integer \( k > 0 \), the graphs of tree-width \( < k \) are well-quasi-ordered by the minor relation. | Null | No |
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. | Yes |
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 12.4.12. (Diestel, Robertson, Seymour & Thomas 1995-99) A graph \( G \) has no \( {K}^{{\aleph }_{0}} \) minor if and only if \( G \) has a tree-decomposition of finite adhesion whose torsos are nearly planar. | Null | No |
Theorem 12.4.13. (Diestel, Robertson, Seymour & Thomas 1992-94) The following assertions are equivalent for connected graphs \( G \) :\n\n(i) \( G \) does not contain \( {K}^{{\aleph }_{0}} \) as a topological minor;\n\n(ii) \( G \) has finite tree-width;\n\n(iii) \( G \) has a normal spanning tree \( T \) such that fo... | Null | No |
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 12.5.2. The Kuratowski set for any minor-closed graph property is finite. | Null | 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... | Yes |
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 2.5. No (non-trivial) knot has an additive inverse. That is, if \( {K}_{1} + {K}_{2} \) is the unknot, then each of \( {K}_{1} \) and \( {K}_{2} \) is unknotted. | Null | No |
Corollary 2.6. If \( K \) is a non-trivial knot and \( \mathop{\sum }\limits_{1}^{n}K \) denotes the sum of \( n \) copies of \( K \), then if \( n \neq m \) it follows that \( \mathop{\sum }\limits_{1}^{n}K \neq \mathop{\sum }\limits_{1}^{m}K \) . There are, then, certainly infinitely many distinct knots. | Null | No |
Corollary 2.7. A knot of genus 1 is prime. | Null | 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 2.9. Schönflies Theorem. Let \( e : {S}^{2} \rightarrow {S}^{3} \) be any piecewise linear embedding. Then \( {S}^{3} - e{S}^{2} \) has two components, the closure of each of which is a piecewise linear ball. | Null | 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 |
Theorem 4.2. Suppose a link \( L \) has an alternating diagram \( D \) . Then \( L \) is a split link if and only if \( D \) is a split diagram. | Null | No |
Theorem 4.4. Suppose \( L \) is a link that has an alternating diagram \( D \) . Then \( L \) is a prime link if and only if \( D \) is a prime diagram. | Null | No |
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 4.6. Suppose that \( L \), with diagram \( D \), is not a split link. Suppose that \( F \) is a 2-sphere meeting \( L \) transversely at two points, with the property that \( F \) separates \( {S}^{3} \) into two 3-balls, neither of which intersects \( L \) in a trivial ball-arc pair. Then \( F \) can be replaced... | Proof. The proof of this lemma follows closely that of the preceding one. In (a), the boundary of the disc \( {\Delta }^{\prime } \) cannot separate, on \( F \), the two points of \( L \cap F \), or else a meridian of \( L \) would be null-homotopic in \( {S}^{3} - L \) . So, \( \partial {\Delta }^{\prime } \) bounds a... | Yes |
Proposition 4.8. Suppose \( L \) is a non-split, prime, alternating link and \( F \) is a closed incompressible surface in \( {S}^{3} - L \) . Then there exists a disc \( \Delta \) spanning \( F \) in \( {S}^{3} \) that meets \( L \) transversely at precisely one point. | Null | No |
Corollary 4.9. Suppose \( L \) is a non-split, prime, alternating link. Any incompressible torus \( T \) contained in \( {S}^{3} - L \) is parallel to the boundary of a solid torus neighbourhood of one of the components of \( L \) . | Null | No |
Proposition 4.11. Suppose \( L \) is a non-split, prime link with alternating diagram D. If \( L \) has a Conway sphere, then it has a Conway sphere \( \sum \) such that \( \sum \cap {S}_{ + } \) is either (i) one curve containing all four points of \( \sum \cap L \) and meeting no saddle, as on the left of Figure 4.7,... | Null | No |
Proposition 5.1. If \( D \) is a link diagram with \( n \) crossings, the Kauffman bracket of \( D \) is given by\n\n\[ \langle D\rangle = \mathop{\sum }\limits_{s}\left( {{A}^{\mathop{\sum }\limits_{{i = 1}}^{n}s\left( i\right) }{\left( -{A}^{-2} - {A}^{2}\right) }^{\left| {sD}\right| - 1}}\right) ,\]\n\nwhere the sum... | Null | No |
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 |
Corollary 5.5. If \( D \) is an adequate diagram, then\n\n\[ M\langle D\rangle - m\langle D\rangle = {2n} + 2\left| {{s}_{ + }D}\right| + 2\left| {{s}_{ - }D}\right| - 4. \] | Null | No |
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 ... | No |
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 \( \mathrm{r}L \) . | No |
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 7.15. Let \( B \) be a path-connected, locally path-connected, semi-locally simply connected space. Then for any subgroup \( G \) of \( {\Pi }_{1}\left( B\right) \), there exists a covering map \( p : {E}_{G} \rightarrow B \), unique up to equivalence, such that \( {p}_{ \star } : {\Pi }_{1}\left( {E}_{G}\right... | Null | No |
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 |
Lemma 8.15. Let \( M \) be a compact orientable 3-manifold such that \( \partial M \) is a connected surface of genus \( g \) . Suppose that \( i : \partial M \rightarrow M \) is the inclusion map. Then the kernel of \( {i}_{ \star } : {H}_{1}\left( {\partial M;\mathbb{Q}}\right) \rightarrow {H}_{1}\left( {M;\mathbb{Q}... | Proof. The following commutative diagram has rows that are parts of the homology and cohomology exact sequences of the pair \( \left( {M,\partial M}\right) \) . Of the vertical arrows, the first and third are Lefschetz duality isomorphisms, and the central one is a Poincaré duality isomorphism.\n\n![5aaec141-7895-41cf-... | 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}... | No |
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 |
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