Signatures of TQFTs and trace fields of two-bridge knots

TL;DR

This paper reveals a surprising connection between the algebraic structures of SO3 TQFTs at specific roots of unity and the trace fields of two-bridge knots, linking quantum topology and knot invariants.

Contribution

It demonstrates that Frobenius algebras from SO3 TQFTs at certain roots encode the trace fields of two-bridge knots, establishing a novel relationship in low-dimensional topology.

Findings

Frobenius algebras at roots of unity contain the trace field of two-bridge knots.

Often, these Frobenius algebras are equal to the trace field.

The work uncovers a new link between TQFT signatures and knot invariants.

Abstract

Let $0<s<r$ be coprime odd integers. We show that the Frobenius algebras governing the signatures of SO$_3$ TQFTs at the root $q=\exp(i\pi s /r)$ contain (and are often equal to) the trace field of the two-bridge knot of parameters $(r,s)$. This gives an intriguing relationship between these two a priori unrelated objects of low-dimensional topology.