Sharp-unknotting number and the Alexander module (Q689943)

Sharp-unknotting is defined using a series of moves in which a pair of strands with the same orientation passes through another such pair. It is known that a finite number of such moves will unknot a given knot; the minimum number required is known as the sharp-unknotting number. The Nakanishi index can be defined as the minimum number of generators of the Alexander module. The paper gives a lower bound for the sharp- unknotting number in terms of the Nakanishi index, and of a similar measure of complexity for the \(p\)-fold cyclic branched cover of the knot; moves relating two knots are also bounded by comparable functions. The results are compared with those for the classical (Gordian) unknotting number, and similarities are noted. By contrast, an infinite sequence of examples of composite knots having sharp-unknotting number 1 is presented.