Jones Polynomial versus Determinant of Quasi-Alternating Links

TL;DR

This paper establishes a finiteness property linking the Jones polynomial and determinant of quasi-alternating links, showing that for a fixed determinant, only finitely many Jones polynomials occur, and vice versa.

Contribution

It proves a new finiteness result connecting the Jones polynomial and determinant for quasi-alternating links, revealing a deep relationship between these invariants.

Findings

Finitely many Jones polynomials for quasi-alternating links of fixed determinant

Finitely many quasi-alternating links for a given Jones polynomial iff for a given determinant

Establishes a bijective finiteness correspondence between Jones polynomial and determinant in this class

Abstract

We prove that there are only finitely many values of the Jones polynomial of quasi-alternating links of a given determinant. Consequently, we prove that there are only finitely many quasi-alternating links of a given Jones polynomial iff there are only finitely many quasi-alternating links of a given determinant.