On alternation numbers of links (Q1041674)

The author constructs infinitely many hyperbolic links whose Gordian distance and \(\tau\)-distance from a given alternating link can be as far as any prescribed positive integer. More precisely, for any \(n \geq 1\), any alternating link \(L\) is concordant to a hyperbolic link \(L_n\) whose Gordian distance and \(\tau\)-distance from \(L\) is \(n\); in fact the minima of these distances (called alternation number and \(\tau\)-alternation number) of \(L_n\) from the set of all alternating links is \(n\). The proof uses an estimate of the \(\tau\)-distance by an Alexander invariant and the topological imitation theory, both were established by the author in earlier works.