Alternating knots with unknotting number one (Q340422)

The author proves that alternating knots with unknotting number 1 have an unknotting crossing in every alternating diagram, thus obtaining a stronger result for alternating knots than that conjectured by \textit{P. Kohn} [Proc. Am. Math. Soc. 113, No. 4, 1135--1147 (1991; Zbl 0734.57007)] for all knots with unknotting number 1. (Kohn conjectured that knots with unknotting number 1 have an unknotting crossing in some minimal diagram.)NEWLINENEWLINEAlso proved is the following which includes a kind of converse to the Montesinos trick for alternating knots with unknotting number 1: A knot is alternating with unknotting number 1 iff its branched double cover can also be obtained by half integer surgery on some knot.NEWLINENEWLINEThe results utilize work of \textit{J. E. Greene} [Ann. Math. (2) 177, No. 2, 449--511 (2013; Zbl 1276.57009); Adv. Math. 255, 672--705 (2014; Zbl 1351.57018)]. This includes the fact that the lattice associated with the Goeritz matrix of an alternating diagram enjoys the ``changemaker'' property.NEWLINENEWLINEA result of Tsukamoto on almost-alternating diagrams of the unknot is reproved and several conjectures are made and questions posed.