Communications in Analysis and Geometry

Volume 26 (2018)

Number 5

Invariants for Turaev genus one links

Pages: 1103 – 1126



Oliver T. Dasbach (Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana, U.S.A.)

Adam M. Lowrance (Department of Mathematics, Vassar College, Poughkeepsie, New York, U.S.A.)


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.

The first author is supported in part by NSF grant DMS-1317942. The second author is supported by Simons Collaboration Grant for Mathematicians no. 355087.

Received 3 May 2016

Published 3 January 2019