For a 3-manifold M , McMullen derived from the Alexander polynomial of M a norm on H(M,R) called the Alexander norm. He showed that the Thurston norm is an upper bound for the Alexander norm. He asked if these two norms were the same when M fibers over the circle. Here, I give examples that show this is not the case. This question relates to the faithfulness of the Gassner representations of th...

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 f...

In this paper, we obtain the existence of almost automorphic solutions to some classes of nonautonomous higher order abstract differential equations with Stepanov almost automorphic forcing terms. A few illustrative examples are discussed at the very end of the paper.

We give a new construction of the one-variable Alexander polynomial of an oriented knot or link, and show that it generalizes to a vector valued invariant of oriented tangles.

We work in the reduced SU(N,K) modular category as constructed recently by Blanchet. We define spin type and cohomological refinements of the Turaev-Viro invariants of closed oriented 3-manifolds and give a formula relating them to Blanchet’s invariants. Roberts’ definition of the Turaev-Viro state sum is exploited. Furthermore, we construct refined Turaev-Viro and Reshetikhin-Turaev TQFT’s and...