Invariants of Montesinos Twins

TL;DR

This paper explores a new relationship between Giller's invariant of Montesinos twins and the Seiberg-Witten invariant, revealing connections between knot invariants and 4-manifold topology.

Contribution

It establishes a link between Giller's invariant for Montesinos twins and the Seiberg-Witten invariant of certain 4-manifold exteriors, extending understanding of knot invariants.

Findings

Giller's invariant relates to the Seiberg-Witten invariant for Montesinos twins.

The invariant is symmetric, unlike the Alexander polynomial.

The work connects knot invariants with 4-manifold topology.

Abstract

C. Giller proposed an invariant of ribbon 2-knots in S^4 based on a type of skein relation for a projection to R^3. In certain cases, this invariant is equal to the Alexander polynomial for the 2-knot. Giller's invariant is, however, a symmetric polynomial -- which the Alexander polynomial of a 2-knot need not be. After modifying a 2-knot into a Montesinos twin in a natural way, we show that Giller's invariant is related to the Seiberg-Witten invariant of the exterior of the twin, glued to the complement of a fiber in E(2).