Signed unknotting number and knot chirality discrimination via strand passage

Abstract

Which chiral knots can be unknotted in a single step by a + to - (+-) crossing change, and which by a - to + (-+) crossing change? Numerical results suggest that if a knot with 6 or fewer crossings can be unknotted by a +- crossing change then it cannot be unknotted by a -+ one, and vice versa. However, we exhibit one chiral 8-crossing knot and one chiral 9-crossing knot which can be unknotted by either crossing change. Furthermore, we address the question analytically using results of Taniyama and Traczyk. We apply Taniyama's classification of unknotting operations to chiral rational knots and fully classify all those which, in a single step, can be unknotted by either type of crossing change; the first of these is 813. As a corollary, we obtain Stoimenow's result that all chiral twist knots can be unknotted by only one of the two crossing change types, +- or -+. Thus, as was observed numerically, all chiral knots with unknotting number one, and seven or fewer crossings, can be unknotted by only one of the two crossing change types. Traczyk's results allow us to address the question for some non-rational chiral unknotting number one knots with 9 or fewer crossings, however, for others the question remains open. We propose a numerical approach for investigating the latter type of knot. We also discuss the implications of our work in the context of DNA topology. Copyright © Progress of Theoretical Physics 2011 All rights reserved.