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Crowell's derived group and twisted polynomials
Daniel S. Silver, Susan G. Williams
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…
An invariant for open virtual strings
Daniel S. Silver, Susan G. Williams
Extended Alexander groups are used to define an invariant for open virtual strings. Examples of non-commuting open strings and a ribbon-concordance obstruction are given. An exampl…
Alexander groups of long virtual knots
Daniel S. Silver, Susan G. Williams
Alexander group systems for virtual long knots are defined and used to show that any virtual knot is the closure of infinitely many long virtual knots. Manturov's result that there…
Lifting representations of Z-groups
Daniel S. Silver, Susan G. Williams
Let K be the kernel of an epimorphism G -> Z, where G is a finitely presented group. If K has infinitely many subgroups of index 2, 3, or 4, then it has uncountably many. Moreover,…