Mathematics – Geometric Topology
Scientific paper
2004-05-20
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.
Anstee Richard P.
Przytycki Jozef H.
Rolfsen Dale
No associations
LandOfFree
Knot polynomials and generalized mutation does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Knot polynomials and generalized mutation, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Knot polynomials and generalized mutation will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-552198