Concordance invariants and the Turaev genus

TL;DR

This paper establishes that differences in various knot concordance invariants provide lower bounds for the Turaev genus, leading to new insights into the structure and classification of quasi-alternating knots.

Contribution

It introduces a novel method linking concordance invariants to Turaev genus, and constructs infinite families of quasi-alternating knots with prescribed Turaev genus.

Findings

Differences in concordance invariants bound the Turaev genus from below.

The additivity of Turaev genus is proven for certain classes of knots.

Constructs infinite families of quasi-alternating knots with fixed Turaev genus.

Abstract

We show that the differences between various concordance invariants of knots, including Rasmussen's $s$-invariant and its generalizations $s_n$-invariants, give lower bounds to the Turaev genus of knots. Using the fact that our bounds are nontrivial for some quasi-alternating knots, we show the additivity of Turaev genus for a certain class of knots. This leads us to the first example of an infinite family of quasi-alternating knots with Turaev genus exactly $g$ for any fixed positive integer $g$, solving a question of Champanerkar-Kofman.