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Mutant knots with symmetry

Published online by Cambridge University Press:  01 January 2009

H. R. MORTON
H. R. MORTON*
Affiliation:
Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL. e-mail: [email protected]
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Abstract

Mutant knots, in the sense of Conway, are known to share the same Homfly polynomial. Their 2-string satellites also share the same Homfly polynomial, but in general their m-string satellites can have different Homfly polynomials for m > 2. We show that, under conditions of extra symmetry on the constituent 2-tangles, the directed m-string satellites of mutants share the same Homfly polynomial for m < 6 in general, and for all choices of m when the satellite is based on a cable knot pattern.

We give examples of mutants with extra symmetry whose Homfly polynomials of some 6-string satellites are different, by comparing their quantum sl(3) invariants.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 1 , January 2009 , pp. 95 - 107
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[1]Aiston, A. K. and Morton, H. R.Idempotents of Hecke algebras of type A. J. Knot Theory Ramifications 7 (1998), 463–487.CrossRefGoogle Scholar
[2]Carbonara, J. O., Remmel, J. B. and Yang, M. Exact formulas for the plethysm s 2[s (1a; b)] and s 12[s (1a; b)]. Technical report, Mathematical Sciences Institute, Cornell University (1992).Google Scholar
[3]Lukac, S. G. Homfly skeins and the Hopf link. PhD. thesis, University of Liverpool (2001).Google Scholar
[4]Morton, H. R. and Cromwell, P. R.Distinguishing mutants by knot polynomials. J. Knot Theory Ramifications 5 (1996), 225–238.CrossRefGoogle Scholar
[5]Morton, H. R. and Manchon, P. M. G. Geometrical relations and plethysms in the Homfly skein of the annulus. Preprint, University of Liverpool (2007).Google Scholar
[6]Morton, H. R. and Ryder, H. J. Mutants and SU(3)q invariants. In ‘Geometry and Topology Monographs’, Vol.1: The Epstein Birthday Schrift. (1998), 365–381.CrossRefGoogle Scholar
[7]Morton, H. R. and Traczyk, P.The Jones polynomial of satellite links around mutants. In ‘Braids’, ed. S. Birman and A. Libgober. Contemp. Math. 78, Amer. Math. Soc. (1988), 587–592.CrossRefGoogle Scholar
[8]Ochiai, M. and Morimura, N. Base tangle decompositions of n-string tangles with 1 < n < 10. Preprint (University of Nara, 2006).Google Scholar
[9]Rosso, M. and Jones, V. F. R.On the invariants of torus knots derived from quantum groups. J. Knot Theory Ramifications 2 (1993), 97–112.CrossRefGoogle Scholar
[10]Stembridge, J. R. A Maple package for symmetric functions. Version 2.4, (2005), University of Michigan, www.math.lsa.umich.edu/~jrs.Google Scholar
[11]Turaev, V. G. Quantum invariants of knots and 3-manifolds. De Gruyter Studies in Mathematics, 18 (Walter de Gruyter and Co., 1994).CrossRefGoogle Scholar