Knot polynomials and generalized mutation

Details Knot polynomials and generalized mutation Knot polynomials and generalized mutation

2004-05-20

arxiv.org/abs/math/0405382v1

Topology and its Applications, 32, 1989, 237-249

Mathematics

Geometric Topology

19 pages, 44 figures; e-print (text and figures) prepared by Dr Makiko Ishiwata

Scientific paper

The motivation for this work was to construct a nontrivial knot with trivial Jones polynomial. Although that open problem has not yielded, the methods are useful for other problems in the theory of knot polynomials. The subject of the present paper is a generalization of Conway's mutation of knots and links. Instead of flipping a 2-strand tangle, one flips a many-string tangle to produce a generalized mutant. In the presence of rotational symmetry in that tangle, the result is called a "rotant". We show that if a rotant is sufficiently simple, then its Jones polynomial agrees with that of the original link. As an application, this provides a method of generating many examples of links with the same Jones polynomial, but different Alexander polynomials. Various other knot polynomials, as well as signature, are also invariant under such moves, if one imposes more stringent conditions upon the symmetries. Applications are also given to polynomials of satellites and symmetric knots.

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