Polynomials of 2-cable-like links

Lickorish, W. B. R.; Lipson, A. S.

doi:10.1090/S0002-9939-1987-0884479-0

Polynomials of $2$-cable-like links
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by W. B. R. Lickorish and A. S. Lipson

Proc. Amer. Math. Soc. 100 (1987), 355-361

DOI: https://doi.org/10.1090/S0002-9939-1987-0884479-0

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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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An invariant of regular isotopy

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