Knot 4--genus and the rank of classes in W(Q(t))

TL;DR

This paper introduces a new, easily computed lower bound on the 4-genus of knots using algebraic invariants derived from Seifert matrices, improving bounds for specific knots.

Contribution

It presents a novel bound on the minimal rank in the Witt group associated with a knot, which is complete modulo torsion and applicable to explicit knot examples.

Findings

New bound on the minimal rank of the class in the Witt group

Bound is complete modulo torsion in the Witt group

Application to specific knots yields new 4-genus bounds

Abstract

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.