New York Journal of Mathematics
Volume 16 (2010) 99-123


Nathan M. Dunfield, Stavros Garoufalidis, Alexander Shumakovitch, and Morwen Thistlethwaite

Behavior of knot invariants under genus 2 mutation

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Published: May 28, 2010
Keywords: mutation, symmetric surfaces, Khovanov Homology, volume, colored Jones polynomial, HOMFLY-PT polynomial, Kauffman polynomial, signature.
Subject: Primary 57N10, Secondary 57M25

Genus 2 mutation is the process of cutting a 3-manifold along an embedded closed genus 2 surface, twisting by the hyper-elliptic involution, and gluing back. This paper compares genus 2 mutation with the better-known Conway mutation in the context of knots in the 3-sphere. Despite the fact that any Conway mutation can be achieved by a sequence of at most two genus 2 mutations, the invariants that are preserved by genus 2 mutation are a proper subset of those preserved by Conway mutation. In particular, while the Alexander and Jones polynomials are preserved by genus 2 mutation, the HOMFLY-PT polynomial is not. In the case of the sl2-Khovanov homology, which may or may not be invariant under Conway mutation, we give an example where genus 2 mutation changes this homology. Finally, using these techniques, we exhibit examples of knots with the same same colored Jones polynomials, HOMFLY-PT polynomial, Kauffman polynomial, signature and volume, but different Khovanov homology.


N.D. was partially supported by the supported by the Sloan Foundation. N.D. and S.G. were partially supported by the U.S. N.S.F.

Author information

Nathan M. Dunfield:
Dept. of Mathematics, MC-382, University of Illinois, Urbana, IL 61801, USA

Stavros Garoufalidis:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA

Alexander Shumakovitch:
George Washington University, Department of Mathematics, 1922 F Street, NW, Washington, DC 20052, USA

Morwen Thistlethwaite:
Department of Mathematics, The University of Tennessee, Knoxville, TN 37996-1300, USA