Crowell's derived group and twisted polynomials

Details Crowell's derived group and twisted polynomials Crowell's derived group and twisted polynomials

2005-06-16

arxiv.org/abs/math/0506339v3

Mathematics

Geometric Topology

16 pages, 6 figures. Version 3 contains new material and extended exposition

Scientific paper

The derived group of a permutation representation, introduced by R.H. Crowell, unites many notions of knot theory. We survey Crowell's construction, and offer new applications. The twisted Alexander group of a knot is defined. Using it, we obtain twisted Alexander modules and polynomials. Also, we extend a well-known theorem of Neuwirth and Stallings giving necessary and sufficient conditions for a knot to be fibered. Virtual Alexander polynomials provide obstructions for a virtual knot that must vanish if the knot has a diagram with an Alexander numbering. The extended group of a virtual knot is defined, and using it a more sensitive obstruction is obtained.

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