Crossing changes and circular Heegaard splittings

TL;DR

This paper connects crossing changes in knots to twists in circular Heegaard splittings, using sutured manifold theory, and suggests an algorithmic approach to detect knot relations based on surface twists.

Contribution

It establishes a relationship between crossing changes and surface twists in circular Heegaard splittings, advancing the understanding of knot transformations and their algorithmic detection.

Findings

Relates genus reducing crossing changes to surface twists in Heegaard splittings.

Proposes an algorithmic method to search for circular Heegaard splittings.

Implications for detecting relations between hyperbolic or fibered knots.

Abstract

We use technology from sutured manifold theory and the theory of Heegaard splittings to relate genus reducing crossing changes on knots in S^3 to twists on surfaces arising in circular Heegaard splittings for knot complements. In a separate paper, currently in preparation, we prove that these circular Heegaard splittings may be searched for algorithmically, and together our results imply that an algorithm to detect when two hyperbolic or fibered knots of different genus are related by a crossing change would follow from an algorithm to determine whether two compact oriented surfaces in S^3 are related by a single twist.