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Mutant knots with symmetry. (English) Zbl 1169.57010
This article considers Homfly polynomials of satellites of pairs of knots that are related by interchanging two tangles, where the tangles that are interchanged satisfy a certain rotational symmetry. If $$K$$ and $$K'$$ are two knots related by such a tangle interchange, it is shown that the $$m$$-string satellites of $$K$$ and $$K'$$ have the same Homfly polynomial when $$m<6$$. (The author provides an example to show that the result does not hold for $$m=6$$.) Furthermore, it is shown that the $$m$$-string connected cables of $$K$$ and $$K'$$ have the same Homfly polynomial for all $$m$$.
It is worth noting that these results can be used to construct mutant knots that are not distinguished by the homfly polynomials of their $$m$$-string satellites for $$m<6$$. This is notable since, in general, mutant knots can be distinguished by the Homfly polynomials of their $$m$$-string satellites for $$m>2$$.

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57M27 Invariants of knots and $$3$$-manifolds (MSC2010)
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##### References:
 [1] DOI: 10.1142/S0218216598000243 · Zbl 0924.57005 · doi:10.1142/S0218216598000243 [2] DOI: 10.1142/S0218216596000163 · Zbl 0866.57002 · doi:10.1142/S0218216596000163 [3] Morton, Contemp. Math. 78 pp 587– (1988) · doi:10.1090/conm/078/975096 [4] DOI: 10.1142/S0218216593000064 · Zbl 0787.57006 · doi:10.1142/S0218216593000064
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