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Proceedings of the American Mathematical Society

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



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.

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

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