State invariants of two-bridge knots

TL;DR

This paper introduces generalized invariants for 2-bridge knots based on Gordon-Litherland bilinear forms, revealing their properties and relating boundary slopes to signature differences.

Contribution

It extends classical invariants like Alexander polynomial and signature to essential state surfaces of 2-bridge knots, providing new insights into their boundary slopes.

Findings

Invariants are well-defined for 2-bridge knots.

Boundary slopes relate to differences in signatures.

Properties of these generalized invariants are characterized.

Abstract

In this paper, we consider generalizations of the Alexander polynomial and signature of 2-bridge knots by considering the Gordon-Litherland bilinear forms associated to essential state surfaces of the 2-bridge knots. We show that the resulting invariants are well-defined and explore properties of these invariants. Finally we realize the boundary slopes of the essential surfaces as a difference of signatures of the knot.