Unknotting operations involving trivial tangles (Q749925)

This paper explores a sequence of operations on knot diagrams, depending on a positive integer n, which involve a simple alteration of a part of the diagram. For each n, repeated use of the operation, called here an H(n)-move, generates an equivalence relation on knots. It is shown here that each knot K is H(n)-equivalent to the unknot, for \(n\geq 2\). The transformation of K to the unknot can be achieved by some minimum number, \(U_ n(K)\) say, of H(n)-moves. The paper gives bounds for \(U_ n(K)\), and also studies bounds for h(K), the least n for which \(U_ n(K)=1\). The H(n)-moves are defined for unoriented diagrams. A similar sequence of moves for oriented diagrams is also discussed, and corresponding results proved.