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Polynomials of $ 2$-cable-like links

Authors: W. B. R. Lickorish and A. S. Lipson
Journal: Proc. Amer. Math. Soc. 100 (1987), 355-361
MSC: Primary 57M25
MathSciNet review: 884479
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Abstract: Morton and Short [MS] have established experimentally that two knots $ {K_1}$ and $ {K_2}$ may have the same $ 2$-variable polynomial $ P(l,m)$ (see [FYHLMO], [LM]) while $ 2$-cables on $ {K_1}$ and $ {K_2}$ can be distinguished by $ P$. We prove here that if $ {K_1}$ and $ {K_2}$ are a mutant pair, then their $ 2$-cables and doubles (and other satellites which are $ 2$-stranded on the boundary of the mutating tangle) cannot be distinguished by $ P$. Similar results are true for the unoriented knot polynomial $ Q$ and its oriented two-variable counterpart $ F$ (see [BLM], [K]). The results are false if $ {K_1},{K_2}$ are links of more than one component.

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

  • [BLM] R. D. Brandt, W. B. R. Lickorish and K. C. Millett, A polynomial invariant for unoriented knots and links, Invent. Math. 84 (1986), 563-573. MR 837528 (87m:57003)
  • [FYHLMO] P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millet and A. Ocneanu, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. (N.S.) 12 (1985), 239-246. MR 776477 (86e:57007)
  • [K] L. H. Kaufmann, An invariant of regular isotopy (to appear).
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  • [MS] H. R. Morton and H. B. Short, The $ 2$-variable polynomial of cable knots (to appear). MR 870598 (88f:57009)
  • [R] D. Rolfsen, Knots and links, Publish or Perish, Berkeley, Calif., 1976. MR 0515288 (58:24236)

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Keywords: Mutant, tangle, $ 2$-stranded satellite, skein generation
Article copyright: © Copyright 1987 American Mathematical Society

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