Invariants for Turaev genus one links
Oliver T. Dasbach, Adam M. Lowrance

TL;DR
This paper characterizes Turaev genus one links by their signature, Jones polynomial coefficients, and all-A state components, revealing structural invariants that distinguish them from other knots.
Contribution
It establishes new invariants for Turaev genus one links, linking signature and Jones polynomial properties to their all-A state configurations.
Findings
Signature determined by all-A state components, positive crossings, and determinant.
Leading or trailing Jones polynomial coefficient has absolute value one for Turaev genus one links.
Turaev genus zero knots are exactly the alternating knots.
Abstract
The Turaev genus defines a natural filtration on knots where Turaev genus zero knots are precisely the alternating knots. We show that the signature of a Turaev genus one knot is determined by the number of components in its all-A Kauffman state, the number of positive crossings, and its determinant. We also show that either the leading or trailing coefficient of the Jones polynomial of a Turaev genus one link (or an almost alternating link) has absolute value one.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
