2005/06/16 by Daniel S. Silver, Susan Williams, Silver, Daniel S. +1 · 3 citations
Mathematics · #20F34 #57M25 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Secondary 20F05
paper · pdf · doi:10.48550/arxiv.math/0506339
openalex publication_date 2005/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.