 

Nathan M. Dunfield, Stavros Garoufalidis, Alexander Shumakovitch, and Morwen Thistlethwaite
Behavior of knot invariants under genus 2 mutation view print erratum


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


Abstract
Genus 2 mutation is the process of cutting a 3manifold along an
embedded closed genus 2 surface, twisting by the hyperelliptic
involution, and gluing back. This paper compares genus 2 mutation
with the betterknown Conway mutation in the context of knots in the
3sphere. 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 HOMFLYPT
polynomial is not. In the case of the sl_{2}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, HOMFLYPT polynomial, Kauffman
polynomial, signature and volume, but different Khovanov homology.


Acknowledgements
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, MC382, University of Illinois, Urbana, IL 61801, USA
nathan@dunfield.info
Stavros Garoufalidis:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 303320160, USA
stavros@math.gatech.edu
Alexander Shumakovitch:
George Washington University, Department of Mathematics, 1922 F Street, NW, Washington, DC 20052, USA
shurik@gwu.edu
Morwen Thistlethwaite:
Department of Mathematics, The University of Tennessee, Knoxville, TN 379961300, USA
morwen@math.utk.edu

