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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク signature of a knot ： ウィキペディア英語版 signature of a knot The signature of a knot is a topological invariant in knot theory. It may be computed from the Seifert surface. Given a knot ''K'' in the 3-sphere, it has a Seifert surface ''S'' whose boundary is ''K''. The Seifert form of ''S'' is the pairing $\backslash phi\; :\; H\_1(S)\; \backslash times\; H\_1(S)\; \backslash to\; \backslash mathbb\; Z$ given by taking the linking number $lk(a^+,b^-)$ where $a,\; b\; \backslash in\; H\_1(S)$ and $a^+,\; b^-$ indicate the translates of ''a'' and ''b'' respectively in the positive and negative directions of the normal bundle to ''S''. Given a basis $b\_1,...,b\_$ for $H\_1(S)$ (where ''g'' is the genus of the surface) the Seifert form can be represented as a ''2g''-by-''2g'' Seifert matrix ''V'', $V\_=\backslash phi(b\_i,b\_j)$. The signature of the matrix $V+V^\backslash perp$, thought of as a symmetric bilinear form, is the signature of the knot ''K''. Slice knots are known to have zero signature. ==The Alexander module formulation== Knot signatures can also be defined in terms of the Alexander module of the knot complement. Let $X$ be the universal abelian cover of the knot complement. Consider the Alexander module to be the first homology group of the universal abelian cover of the knot complement: $H\_1(X;\backslash mathbb\; Q)$. Given a $\backslash mathbb\; Q(Z)$-module $V$, let $\backslash overline$ denote the $\backslash mathbb\; Q(Z)$-module whose underlying $\backslash mathbb\; Q$-module is $V$ but where $\backslash mathbb\; Z$ acts by the inverse covering transformation. Blanchfield's formulation of Poincaré duality for $X$ gives a canonical isomorphism $H\_1(X;\backslash mathbb\; Q)\; \backslash simeq\; \backslash overline$ where $H^2(X;\backslash mathbb\; Q)$ denotes the 2nd cohomology group of $X$ with compact supports and coefficients in $\backslash mathbb\; Q$. The universal coefficient theorem for $H^2(X;\backslash mathbb\; Q)$ gives a canonical isomorphism with $Ext\_(H\_1(X;\backslash mathbb\; Q),\backslash mathbb\; Q(Z))$ (because the Alexander module is $\backslash mathbb\; Q(Z)$-torsion). Moreover, just like in the quadratic form formulation of Poincaré duality, there is a canonical isomorphism of $\backslash mathbb\; Q(Z)$-modules $Ext\_(H\_1(X;\backslash mathbb\; Q),\backslash mathbb\; Q(Z))\; \backslash simeq\; Hom\_(H\_1(X;\backslash mathbb\; Q),\; )$, where $$. This isomorphism can be thought of as a sesquilinear duality pairing $H\_1(X;\backslash mathbb\; Q)\; \backslash times\; H\_1(X;\backslash mathbb\; Q)\; \backslash to$ where $$. This form takes value in the rational polynomials whose denominators are the Alexander polynomial of the knot, which as a $\backslash mathbb\; Q(Z)$-module is isomorphic to $\backslash mathbb\; Q(Z)/\backslash Delta\; K$. Let $tr\; :\; \backslash mathbb\; Q(Z)/\backslash Delta\; K\; \backslash to\; \backslash mathbb\; Q$ be any linear function which is invariant under the involution $t\; \backslash longmapsto\; t^$, then composing it with the sesquilinear duality pairing gives a symmetric bilinear form on $H\_1\; (X;\backslash mathbb\; Q)$ whose signature is an invariant of the knot. All such signatures are concordance invariants, so all signatures of slice knots are zero. The sesquilinear duality pairing respects the prime-power decomposition of $H\_1\; (X;\backslash mathbb\; Q)$—i.e.: the prime power decomposition gives an orthogonal decomposition of $H\_1\; (X;\backslash mathbb\; R)$. Cherry Kearton has shown how to compute the ''Milnor signature invariants'' from this pairing, which are equivalent to the ''Tristram-Levine invariant''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「signature of a knot」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.