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Colored HOMFLY-PT polynomials that distinguish mutant knots. (English) Zbl 1405.57017
Let $$K$$ be a knot in $$S^3$$, and let $$S$$ be a sphere that intersects $$K$$ transversely in four points, dividing it into two 2-string tangles. Let $$K'$$ be the knot obtained by cutting along $$S$$, rotating one of the tangles $$180^\circ$$ along one of the axes $$x$$, $$y$$, or $$z$$, and gluing it back. $$K$$ and $$K'$$ are called a mutant pair. It is known that many polynomial invariants, like the Jones, HOMFLY-PT and Kauffman polynomials do not distinguish between pairs of mutant knots. In this paper it is explained how certain quantum colored HOMFLY-PT polynomials with multiplicity structure can distinguish between mutant knots. In particular, an explicit calculation is made of a colored HOMPLY-PT polynomial of the Kinoshita-Terasaka and Conway knots, a famous pair of mutant knots, showing that they have different polynomials.

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57R56 Topological quantum field theories (aspects of differential topology)
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