Genus generators and the positivity of the signature

TL;DR

This paper explores the relationship between the signature of positive links and their genus, providing an algorithmic approach to analyze genus bounds and their relation to knot complexity.

Contribution

It introduces a generator-based method for canonical genus and demonstrates how to partially decide the boundedness of positive knots' genera with fixed signature.

Findings

The set of knots with genus ≥ n is finitely dominated.

Algorithmic partial decision procedure for genus bounds.

Relation established between genus, signature, and Taniyama's order.

Abstract

It is a conjecture that the signature of a positive link is bounded below by an increasing function of its negated Euler characteristic. In relation to this conjecture, we apply the generator description for canonical genus to show that the boundedness of the genera of positive knots with given signature can be algorithmically partially decided. We relate this to the result that the set of knots of canonical genus greater than or equal to n is dominated by a finite subset of itself in the sense of Taniyama's partial order.