Invariants for Turaev genus one links

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.

17 pages, 13 figures. Theorem 1.3 in version 1 is now Equation 3.2 in version 2. To appear in Communications in Analysis and Geometry

References in corpus (2)

• The Jones polynomial and graphs on surfaces
• Almost alternating knots with 12 crossings and Turaev genus

Cited by in corpus (5)

• Extremal Khovanov homology of Turaev genus one links
• Jones slopes and coarse volume of near-alternating links
• Near extremal Khovanov homology of Turaev genus one links
• The Jones polynomial of an almost alternating link
• On the Turaev genus of torus knots