Alexander Module Research Articles

On the nullity and enclosure genus of wild knots

The fundamental group of the complement of a wild knot in a 3-sphere can be expressed as the colimit (direct limit) of a suitable family of groups and homomorphisms (Crowell [4]). To each group in the family we assign a Jacobian module, and in ?1 we prove that this assignment is functorial and preserves colimits. This is used in ?2 to show that the nullity of the Alexander module of a knot with one wild point is bounded above by its enclosure genus. This can be used in some cases to calculate the enclosure genus and the penetration index of the knot in a purely algebraic way. In ?3 we give examples to show that the upper bound of Theorem 2 is the best possible, along with some results concerning the penetration index of an arc relative to surfaces of genus r.

Pretzel knots with unknotting number one

We provide a partial classification of the 3-strand pretzel knots $K = P(p,q,r)$ with unknotting number one. Following the classification by Kobayashi and Scharlemann-Thompson for all parameters odd, we treat the remaining families with $r$ even. We discover that there are only four possible subfamilies which may satisfy $u(K) = 1$. These families are determined by the sum $p+q$ and their signature, and we resolve the problem in two of these cases. Ingredients in our proofs include Donaldson's diagonalisation theorem (as applied by Greene), Nakanishi's unknotting bounds from the Alexander module, and the correction terms introduced by Ozsv\'ath and Szab\'o. Based on our results and the fact that the 2-bridge knots with unknotting number one are already classified, we conjecture that the only 3-strand pretzel knots $P(p,q,r)$ with unknotting number one that are not 2-bridge knots are $P(3,-3,2)$ and its reflection.