Upper bounds in the Ohtsuki-Riley-Sakuma partial order on 2-Bridge knots
Work of \textit{T. Ohtsuki, R. Riley} and \textit{M. Sakuma} [in: M. Boileau (ed.) et al., The Zieschang Gedenkschrift. Coventry: Geometry \& Topology Publications. Geometry and Topology Monographs 14, 417--450 (2008; Zbl 1146.57011)] that relates to the existence of certain epimorphisms between the complements of 2-bridge links can be used to define a partial order on 2-bridge knots. A refinement of this partial order is given by that of \textit{D. Silver} and \textit{W. Whitten} [J. Knot Theory Ramifications 15, 153--166 (2006; Zbl 1096.57007)].NEWLINENEWLINEA key concept in defining the Ohtsuki-Riley-Sakuma partial order, and in the paper under review, is a particular normalisation of the continued fraction expansion of a rational number defining a given 2-bridge knot. The rational number is expressed as an expanded even vector. That is, it is expressed as a finite sequence from \(\{2,0,-2\}\), such that whenever a 0 occurs the components on either side are either both 2 or both \(-2\). The partial order is given by comparing the expanded even vectors of the knots in question.NEWLINENEWLINEThis paper considers the question of finding an upper bound on a set of 2-bridge knots. Any such set with an upper bound is finite. It is shown that a finite set of 2-bridge knots has an upper bound if and only if every two-element subset has an upper bound. A necessary and sufficient condition is given for the existence of an upper bound for two incomparable 2-bridge knots. The proof is constructive.NEWLINENEWLINEIn addition, given two 2-bridge knots, a graphical interpretation is given of an expanded even vector representing an upper bound on the knots that is of minimal length among all such vectors.
scientific article; zbMATH DE number 6044374
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