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 Kaufiman state, the number of positive crossings, and its determinant. We also show that either the leading or trailing coeficient of the Jones polynomial of a Turaev genus one link (or an almost alternating link) has absolute value one.
Publication Source (Journal or Book title)
Communications in Analysis and Geometry
Dasbach, O., & Lowrance, A. (2018). Invariants for turaev genus one links. Communications in Analysis and Geometry, 26 (5), 1103-1126. https://doi.org/10.4310/CAG.2018.v26.n5.a4