A new partial ordering of knots (Q2348588)

Recall the partial ordering of knots defined by K. Taniyama to be \(K_1 \leq K_2\) if every diagram of \(K_2\) can be transformed into some diagram of \(K_1\) via simultaneous crossing changes. In this paper the authors define a new ordering of knots, called \(V\)-order, to be \(K_1 \leq K_2\) if there is a minimal diagram of \(K_2\) that can be transformed into some diagram of \(K_1\) via simultaneous crossing changes. In order for the reflexive relation for the \(V\)-order to be a partial ordering, we need to restrict to prime alternating knots, since in the \(V\)-order, there are non-alternating knots \(K_1\) and \(K_2\) with \(K_1 \leq K_2\) and \(K_2 \leq K_1\) but \(K_1 \neq K_2\). It is proved that, in the \(V\)-order for prime alternating knots, if \(K_1 \leq K_2\), then the crossing numbers of \(K_1\) and \(K_2\) satisfy \(C(K_1) \leq C(K_2)\), the bridge numbers of \(K_1\) and \(K_2\) satisfy \(br(K_1) \leq br(K_2)\), and the braid indices of \(K_1\) and \(K_2\) satisfy \(b(K_1) \leq b(K_2)\). The authors also investigate the relation of immediate predecessor, called direct \(V\)-minor, on the class of prezel knots, and conjecture that they are the only prime alternating knots with only one direct minor.