Unknotting Numbers and Minimal Knot Diagrams

Abstract

We show there is an infinite number of knots whose unknotting numbers can only be realized through a sequence of crossing changes on a nonminimal projection, if ambient isotopies between crossing changes are not allowed. In this process, the question arises one crossing change at a time is made, always on a minimal projection, in order to obtain the unknotting number of a knot.

A Partial Ordering of Knots and Links Through Diagrammatic Unknotting

“…In this section, we will discuss a concept that can be thought of as a generalization of the unknotting number restricted to minimal diagrams of a knot introduced and studied in [3] and [20]. Definition 4.1.…”

“…The concept of u D was first discussed in [3], though worded differently. Recall that a knot K has unknotting number u(K) = n if there exists a projection of the knot such that changing n crossings in the projection turns the knot projection into the unknot and no projection of K exists such that changing fewer than n crossings would do this.…”

On unknotting numbers and knot trivadjacency

“…Then for the calculation of u one still has to consider only finitely many diagrams, and one hopes for Conjecture 3.2 (see [8], [21]). u(K) = u (K) for any knot K.…”

“…If k = 2, then p = 5, so assume k > 2. Also 2l ≤ p + 1 and 2k ≤ p + 1 from the middle condition in (8).…”

“…Now k ≥ 3 leads to p ≤ k + 2, which together with the obvious k ≤ p and p ≡ ±1 (k) leads to k = 3 and again p = 5, or to p = k + 1 and a contradiction to the middle condition in (8).…”