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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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