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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 Arf invariant of a knot ： ウィキペディア英語版 Arf invariant of a knot In the mathematical field of knot theory, the Arf invariant of a knot, named after Cahit Arf, is a knot invariant obtained from a quadratic form associated to a Seifert surface. If ''F'' is a Seifert surface of a knot, then the homology group H1(''F'', Z/2Z) has a quadratic form whose value is the number of full twists mod 2 in a neighborhood of an imbedded circle representing an element of the homology group. The Arf invariant of this quadratic form is the Arf invariant of the knot. ==Definition by Seifert matrix== Let $V\; =\; v\_$ be a Seifert matrix of the knot, constructed from a set of curves on a Seifert surface of genus ''g'' which represent a basis for the first homology of the surface. This means that ''V'' is a 2''g'' × 2''g'' matrix with the property that ''V'' − ''V''T is a symplectic matrix. The ''Arf invariant'' of the knot is the residue of :$\backslash sum\backslash limits^g\_v\_v\_\; \backslash pmod\; 2.$ Specifically, if $\backslash ,\; i=1...g$, is a symplectic basis for the intersection form on the Seifert surface, then :$Arf(K)\; =\; \backslash sum\backslash limits^g\_lk(a\_i,\; a\_i^)lk(b\_i,b\_i^)\; \backslash pmod\; 2.$ where $a^+$ denotes the positive pushoff of ''a''. 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「Arf invariant 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.