A novel connection between integral binary quadratic forms and knot polynomials

TL;DR

This paper uncovers a new link between algebraic number theory and knot theory, showing that class numbers of quadratic fields correspond to classes of links with specific invariants, revealing limitations of classical knot polynomials.

Contribution

It establishes a novel algebraic correspondence between binary quadratic forms and links of braid index at most three, connecting class numbers to link invariants.

Findings

Number of quadratic form classes equals number of certain link isotopy classes.

Class numbers measure the failure of Alexander/Jones polynomials to distinguish links.

Links with prescribed invariants correspond to quadratic forms of specific discriminants.

Abstract

We establish a novel connection between algebraic number theory and knot theory. We show that the number of equivalence classes of integral binary quadratic forms of discriminant $t^2 - 4$ (for $t\neq \pm 2$) is equal to the number of isotopy classes of links in $\mathbb{S}^3$ with prescribed values (depending on $t$) of three classical link invariants. The equality arises from a natural algebraic correspondence between integral binary quadratic forms (of discriminant $t^2 - 4$ for $t\neq \pm 2$) and isotopy classes of links of braid index at most three. In particular, the class numbers of certain quadratic number fields precisely measure the failure of the Alexander/Jones polynomial to distinguish non-isotopic links of braid index at most three.