Alexander polynomials of ribbon knots and virtual knots

TL;DR

This paper establishes a new formula for the Alexander polynomial of ribbon knots based on intrinsic singularity data, introduces a new invariant called half Alexander polynomial, and extends these ideas to virtual knots.

Contribution

It introduces the half Alexander polynomial as a new invariant, provides formulas to compute it, and characterizes its possible forms, also deriving new formulas for virtual knots.

Findings

Alexander polynomial of a ribbon knot is determined by its singularity data.

Defined the half Alexander polynomial as an invariant of ribbons.

Derived new formulas for Alexander polynomials of virtual knots.

Abstract

We find that Alexander polynomial of a ribbon knot in $ \mathbb{Z}HS^3 $ is determined by the intrinsic singularity information of its ribbon, and give a formula to calculate Alexander polynomial of a ribbon knot by that. We define half Alexander polynomial $ A_R (t) $, an invariant of oriented ribbons, and in fact the Alexander polynomial of the ribbon knot is $ A_R (t) A_R (t^{-1}) $. We give two useful simplified formulas for half Alexander polynomial. We characterize completely the polynomials arising as half Alexander polynomials of ribbons. The above study unexpectedly leads us to discover new formulas for Alexander polynomial of general knots and virtual knots in terms of Gauss diagrams.