K-stable equivalence for knots in Heegaard surfaces

TL;DR

This paper introduces K-stable equivalence for knots in Heegaard surfaces and proves that pairs with the same surface slope are K-stably equivalent, providing a new classification tool in 3-manifold topology.

Contribution

It defines K-stable equivalence for knots in Heegaard surfaces and establishes its equivalence criterion based on surface slope.

Findings

K-stable equivalence is well-defined for knots in Heegaard surfaces.

Pairs with the same surface slope are K-stably equivalent.

Provides a new perspective for classifying knots in 3-manifolds.

Abstract

Let K be a knot embedded in a Heegaard surface S for a closed orientable 3-manifold M. We define K-stable equivalence between pairs (S, K) and (S', K) in M, and we prove that any two pairs are K-stably equivalent in M if they have the same surface slope.