Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Mutations of links in genus 2 handlebodies

Authors: D. Cooper and W. B. R. Lickorish
Journal: Proc. Amer. Math. Soc. 127 (1999), 309-314
MSC (1991): Primary 57M25; Secondary 81T99, 81R50
MathSciNet review: 1605940
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A short proof is given to show that a link in the 3-sphere and any link related to it by genus 2 mutation have the same Alexander polynomial. This verifies a deduction from the solution to the Melvin-Morton conjecture. The proof here extends to show that the link signatures are likewise the same and that these results extend to links in a homology 3-sphere.

References [Enhancements On Off] (What's this?)

  • 1. D.Bar-Natan and S.Garoufalidis, On the Melvin-Morton-Rozansky conjecture, Invent. Math. 125 (1996), 103-133. MR 97i:57004
  • 2. J.Dean, Many classical knot invariants are not Vassiliev invariants, J. Knot Theory and its Ramifications 3 (1994), 7-10. MR 94k:57008
  • 3. A.Kawauchi, Topological imitation, mutation and quantum SU($2$) invariant, J. Knot Theory and its Ramifications 3 (1994), 25-39. MR 95a:57025
  • 4. P.M.Melvin and H.R.Morton, The coloured Jones function, Comm. Math. Phys. 169 (1995), 501-520. MR 96g:57012
  • 5. H.R.Morton and H.B.Short, Calculating the $2$-variable polynomial for knots presented as closed braids, J. Algorithms 11 (1990), 117-131. MR 91f:57004
  • 6. K.Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965), 387-422. MR 30:1506
  • 7. L.Rozansky, A contribution of the trivial connection to the Jones polynomial and Witten's invariant of 3d manifolds, I and II, Comm. Math. Phys. 2 175 (1996), 275-296, 297-318. MR 97e:57038
  • 8. R.Trapp, Twist sequences and Vassiliev invariants, J. Knot Theory and its Ramifications 3 (1994), 391-405. MR 95h:57012
  • 9. A.G.Tristram, Some cobordism invariants for links, Proc. Cambridge Philos. Soc. 66 (1969), 251-264. MR 40:2104

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 57M25, 81T99, 81R50

Retrieve articles in all journals with MSC (1991): 57M25, 81T99, 81R50

Additional Information

D. Cooper
Affiliation: Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California 93106

W. B. R. Lickorish
Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge, CB2 1SB, United Kingdom

Keywords: Alexander polynomial, knot signature, knot mutation, Jones polynomial, Melvin-Morton conjecture
Received by editor(s): May 13, 1997
Additional Notes: This research was supported in part by N.S.F. grants DMS9504438 and DMS9510505.
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society