Hostname: page-component-7c8c6479df-xxrs7 Total loading time: 0 Render date: 2024-03-29T08:46:37.580Z Has data issue: false hasContentIssue false

The 2-variable polynomial of cable knots

Published online by Cambridge University Press:  24 October 2008

H. R. Morton
Affiliation:
Department of Pure Mathematics, University of Liverpool, Liverpool L69 3BX
H. B. Short
Affiliation:
Department of Pure Mathematics, University of Liverpool, Liverpool L69 3BX

Abstract

The 2-variable polynomial PK of a satellite K is shown not to satisfy any formula, relating it to the polynomial of its companion and of the pattern, which is at all similar to the formulae for Alexander polynomials. Examples are given of various pairs of knots which can be distinguished by calculating P for 2-strand cables about them even though the knots themselves share the same P. Properties of a given knot such as braid index and amphicheirality, which may not be apparent from the knot's polynomial P, are shown in certain cases to be detectable from the polynomial of a 2-cable about the knot.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1987

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Birman, J. S., On the Jones polynomial of closed 3-braids. Invent. Math 81 (1985), 287294.CrossRefGoogle Scholar
[2]Fox, R. H.. Free differential calculus V. The Alexander matrices re-examined. Ann. Math. 71 (1960), 408422.CrossRefGoogle Scholar
[3]Franks, J. and Williams, R. F.. Braids and the Jones-Conway polynomial. Preprint, Northwestern University, 1985.Google Scholar
[4]Franks, J. and Williams, R. F.. Braids and the Jones-Conway polynomial. Abstracts Amer. Math. Soc. 6 (1985), 355.Google Scholar
[5]Freyd, P., Yetter, D., Hoste, J., Lickorish, W. B. R., Millett, K. C. and Ocneanu, A., A new polynomial invariant of knots and links. Bull. Amer. Math. Soc. 12 (1985), 239246.CrossRefGoogle Scholar
[6]Jones, V. F. R.. An introduction to Hecke algebras. Seminar notes, M.S.R.I. 1984.Google Scholar
[7]Jones, V. F. R.. Private communication.Google Scholar
[8]Kauffman, L.. TwO two-variable polynomials. Preprint, 1985.Google Scholar
[9]Lickorish, W. B. R. and Millett, K. C.. A polynomial invariant of oriented links. Preprint, 1985. To appear in Topology.Google Scholar
[10]Morton, H. R.. Seifert circles and knot polynomials. Math. Proc. Cambridge Philos. Soc. 99 (1986), 107109.CrossRefGoogle Scholar
[11]Morton, H. R.. Exchangeable braids. In Low dimensional topology, LMS lecture notes 95 (Cambridge University Press, 1985), 86105.CrossRefGoogle Scholar
[12]Morton, H. R. and Short, H. B.. Calculating the 2-variable polynomial for knots presented as closed braids. Preprint, Liverpool University 1986.Google Scholar
[13]Ocneanu, A.. A polynomial invariant for knots; a combinatorial and an algebraic approach. Preprint, M.S.R.I. 1985.Google Scholar
[14]Thistlethwaite, M. B.. Tables of knot polynomials. Polytechnic of the South Bank, London, 1985.Google Scholar
[15]Lickorish, W. B. R. and Lipson, A. S.. Polynomials of 2-cable-like links. Preprint, Cambridge University, 1986.CrossRefGoogle Scholar