Communications in Analysis and Geometry

Volume 26 (2018)

Number 5

Invariants for Turaev genus one links

Pages: 1103 – 1126

DOI: https://dx.doi.org/10.4310/CAG.2018.v26.n5.a4

Authors

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.)

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.

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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