Knot 4--genus and the rank of classes in W(Q(t))
Knot 4--genus and the rank of classes in W(Q(t))
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结4--W(Q(t))中的属和类的等级
DOI：
10.2140/pjm.2011.252.113
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发表时间：
2009
期刊：
影响因子：
--
通讯作者：
C. Livingston
中科院分区：
文献类型：
--
作者：
C. Livingston
对于结点K的Seifert矩阵，可以将矩阵w(K)与有理函数域Q(T)中的项联系起来。Murasugi、Milnor和Levine-Tristram纽结签名都提供了纽结的4-亏格上的界，由w(K)确定。更一般地，由w(K)表示的类的代表在Q(T)上的Hermitian形式的Witt群中的最小秩为K的4-亏格提供了一个下界。这里我们描述了由w(K)表示的类的最小秩上的一个容易计算的新的界。此外，这个下界在Witt群中是完全模挠率。具体地说，如果秩上的界是M，则4w(K)有一个秩正好是4M的代表。给出了显式纽结的应用，找到了通过其他方法无法获得的特定纽结的4亏格界限。
To a Seifert matrix of a knot K one can associate a matrix w(K) with entries in the rational function field, Q(t). The Murasugi, Milnor, and Levine-Tristram knot signatures, all of which provide bounds on the 4-genus of a knot, are determined by w(K). More generally, the minimal rank of a representative of the class represented by w(K) in the Witt group of hermitian forms over Q(t) provides a lower bound for the 4-genus of K. Here we describe an easily computed new bound on the minimal rank of the class represented by w(K). Furthermore, this lower bound is complete modulo torsion in the Witt group. Specifically, if the bound on the rank is M, then 4w(K) has a representative of rank exactly 4M. Applications to explicit knots are given, finding 4-genus bounds for specific knots that are unattainable via other approaches.