Alternating knots with unknotting number one

We prove that if an alternating knot has unknotting number one, then there exists an unknotting crossing in any alternating diagram. This is done by showing that the obstruction to unknotting number one developed by Greene in his work on alternating 3-braid knots is sufficient to identify all unknotting number one alternating knots. As a consequence, we also get a converse to the Montesinos trick: an alternating knot has unknotting number one if its branched double cover arises as half-integer surgery on a knot in $S^3$. We also reprove a characterisation of almost-alternating diagrams of the unknot originally due to Tsukamoto.Comment: 38 pages, 15 figures. This a significant revision of the first version. A proof of Tsukamoto's work on almost-alternating diagrams of the unknot is now included. There is also an additional formulation of the main theorem which makes precise the signs of the unknotting crossings and the resulting half-integer surgeries. Some comments on potential further questions have also been adde

On calculating the slice genera of 11- and 12-crossing knots

This paper contains the results of efforts to determine values of the smooth and the topological slice genus of 11- and 12-crossing knots. Upper bounds for these genera were produced by using a computer to search for genus one concordances between knots. For the topological slice genus further upper bounds were produced using the algebraic genus. Lower bounds were obtained using a new obstruction from the Seifert form and by use of Donaldson's diagonalization theorem. These results complete the computation of the topological slice genera for all knots with at most 11 crossings and leaves the smooth genera unknown for only two 11-crossing knots. For 12 crossings there remain merely 25 knots whose smooth or topological slice genus is unknown.Comment: 9 pages + 11 pages of appendices. This is a substantial expansion of the original article. This version features a second author and new techniques for calculating the topological slice genu

Characterizing slopes for the (−2,3,7)(-2,3,7)-pretzel knot

In this note we exhibit concrete examples of characterizing slopes for the knot $12n242$, aka the $(-2,3,7)$-pretzel knot. Although it was shown by Lackenby that every knot admits infinitely many characterizing slopes, the non-constructive nature of the proof means that there are very few hyperbolic knots for which explicit examples of characterizing slopes are known.Comment: 9 page