Mock Seifert matrices and unoriented algebraic concordance

TL;DR

This paper introduces mock Seifert matrices to define a new unoriented algebraic concordance group for knots in thickened surfaces, providing new invariants and insights into virtual knot concordance and related topological properties.

Contribution

It develops the concept of mock Seifert matrices, defines the unoriented algebraic concordance group, and introduces new invariants for virtual knots, expanding the algebraic tools in knot theory.

Findings

The group al G^Z is infinitely generated and contains a large subgroup.

Mock Seifert matrices lead to new invariants like the mock Alexander polynomial.

Applications include results on virtual knot concordance, crosscap numbers, and Seifert genus.

Abstract

A mock Seifert matrix is an integral square matrix representing the Gordon-Litherland form of a pair $(K,F)$, where $K$ is a knot in a thickened surface and $F$ is an unoriented spanning surface for $K$. Using these matrices, we introduce a new notion of unoriented algebraic concordance, as well as a new group denoted $\mathcal{m} \mathcal{G}^{\mathbb Z}$ and called the unoriented algebraic concordance group. This group is abelian and infinitely generated. There is a surjection $\lambda \colon \mathcal{v} \mathcal{C} \to \mathcal{m} \mathcal{G}^{\mathbb Z}$, where $\mathcal{v} \mathcal{C} $ denotes the virtual knot concordance group. Mock Seifert matrices can also be used to define new invariants, such as the mock Alexander polynomial and mock Levine-Tristram signatures. These invariants are applied to questions about virtual knot concordance, crosscap numbers, and Seifert genus for knots in thickened surfaces. For example, we show that $\mathcal{m} \mathcal{G}^{\mathbb Z}$ contains a copy of ${\mathbb Z}^\infty \oplus ({\mathbb Z}/2)^\infty \oplus({\mathbb Z}/4)^\infty.$