Chord index for knots in thickened surfaces

TL;DR

This paper introduces a chord index homomorphism for knots in thickened surfaces, providing a new algebraic tool to study knot invariants in this setting.

Contribution

It constructs a novel chord index homomorphism from a subgroup of surface homology to chord indices of knots, expanding the algebraic framework for knot invariants.

Findings

Defines a chord index homomorphism for knots in thickened surfaces

Discusses knot invariants derived from the homomorphism

Provides new algebraic methods for analyzing knots in surfaces

Abstract

In this note, we construct a chord index homomorphism from a subgroup of $H_1(\Sigma, \mathbb{Z})$ to the group of chord indices of a knot $K$ in $\Sigma\times I$. Some knot invariants derived from this homomorphism are discussed.