Signs of knot polynomial evaluations from a topological perspective

TL;DR

This paper demonstrates that certain knot polynomial evaluations are uniquely determined by the topology of their double branched covers and provides explicit formulas and bounds related to knot invariants.

Contribution

It establishes the topological determination of specific polynomial evaluations and introduces explicit formulas and new bounds for unknotting numbers.

Findings

Evaluation of Jones polynomial at sixth root of unity is topologically determined.

Evaluation of Q-polynomial at reciprocal of golden ratio is topologically determined.

New lower bounds for unknotting numbers of knots and links are derived.

Abstract

We prove that for knots, the evaluation of the Jones polynomial at the sixth root of unity, as well as the evaluation of the $Q$-polynomial at the reciprocal of the golden ratio, are uniquely determined by the oriented homeomorphism type of the double branched covering. We provide explicit formulae for these evaluations in terms of the linking pairing. The proof proceeds via so-called singular determinants, from which we also extract new lower bounds for the unknotting numbers of knots and links.