Gordian adjacency for torus knots
An unknotting sequence for a knot \(K\) is a sequence \(K=K_0,\ldots,K_n\) of knots such that \(K_n\) is the unknot, \(K_i\) differs from \(K_{i-1}\) by a single crossing change for each \(i\), and \(n\) is minimal among such sequences. Then \(K\) is said to have unknotting number \(n\). A second knot \(K'\) is said to be Gordian adjacent to \(K\) if it occurs in some unknotting sequence for \(K\). Note that Gordian adjacency gives a partial order on knots, and is not symmetric. This paper addresses the question of when one torus knot is Gordian adjacent to another. The first result shows that reducing one or both of the defining integers gives Gordian adjacent torus knots. The second specifies when an index 2 torus knot is Gordian adjacent to an index 3 torus knot (the index of a torus knot is the smaller of the two defining integers). Next the author considers the asymptotics of the values of torus knots that are Gordian adjacent, and the final section discusses the relationship between Gordian adjacency and another partial order called algebraic adjacency. The positive results about adjacency are shown by a combination of direct construction and studying isotopies of knots in a thickened torus. The negative results are shown by using Levine--Tristam signatures as an obstruction to Gordian adjacency.
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