Invariants of knot diagrams

Abstract

We construct a new order 1 invariant for knot diagrams. We use it to determine the minimal number of Reidemeister moves needed to pass between certain pairs of knot diagrams.

Unknot Diagrams Requiring a Quadratic Number of Reidemeister Moves to Untangle

“…In [5] it is shown that the change in the value of I lk resulting from a Reidemeister move has one of the following forms:…”

“…We recall the definition of the invariant of knot diagrams in S 2 introduced in [5]. We denote the set of all knot diagrams in S 2 by D and the set of all two component links in R 3 by L. Given a knot diagram D ∈ D and a crossing a in D, define D a ∈ L to be the two-component link obtained by smoothing the crossing a (respecting the orientation of the strands).…”

“…Our main tool is an invariant of knot diagrams developed in [5] and used there to obtain new linear lower bounds on the Reidemeister distance.…”

Minimal sequences of Reidemeister moves on diagrams of torus knots

“…We let b i denote σ This theorem can also be obtained by using Arnold invariants. Hass and Nowik showed in section 4 in [6] that any lower bound of the number of Reidemeister moves between two knot diagrams obtained from the cowrithe coincides with that from the sum of adequately normalized Arnold invariants St + J + /2 of the underlying spherical closed curve.…”

“…In [3], Carter, Elhamdadi, Saito and Satoh gave a lower bound for the number of RIII moves by using extended n-colorings of knot diagrams in R 2 . Hass and Nowik introduced a certain knot diagram invariant by using smoothing and linking numbers in [6], and gave in [7] an example of an infinite sequence of diagrams of the trivial knot such that the n-th one has 7n − 1 crossings, can be unknotted by 2n…”