A strand passage metric for topoisomerase action (Q2708754)

Let \(u(K_1,K_2)\) denote the distance between two knots. Quoting from the introduction: ``By doing the crossing changes on a particular diagram, either by hand or by computer, one can easily determine upper bounds for the distance between two given knots. The difficult question is then knowing whether or not there exists a shorter path. For this lower bounds are needed. Murakami generalized Murasugi's signature lower bound for the unknotting number to obtain the following lower bound for the strand passage metric: NEWLINE\[NEWLINEu(K_1,K_2)\geq\textstyle{1\over 2}\bigl|\sigma(K_1)-\sigma(K_2)\bigr|NEWLINE\]NEWLINE This lower bound also holds for semi-oriented links. Theorem 1 gives a determination of when the distance between two unoriented 2-bridge knots or links is one and can be used to find distance one 2-bridge knots (links) where the crossing change does not occur in a minimal diagram. It is a generalization of \textit{T. Kanenobu} and \textit{H. Murakami}'s result for the unknot [Proc. Am. Math. Soc. 98, 499-502 (1986; Zbl 0613.57002)] and \textit{P. Kohn}'s result for the unlink of two components [ibid. 113, No. 4, 1135-1147 (1991; Zbl 0734.57007)]. Proposition 2 gives the first homology group of knots of distance one from a particular 4-plat and is a generalization of \textit{P. Kohn}'s proposition 2 in [Osaka J. Math. 30, No. 4, 741-752 (1993; Zbl 0822.57003)]. A table for the strand passage metric for knots is generated by computer''.NEWLINENEWLINEFor the entire collection see [Zbl 0959.57001].