Narrow Results

In the classical knot theory there is a well-known notion of descending diagram. From an arbitrary diagram one can easily obtain, by some crossing changes, a descending diagram which is a diagram of the unknot or unlink. In this paper the notion of descending diagram for knots and links in ℝ³ is extended to the case of nonoriented knots and links in the projective space. It is also shown that this notion cannot be extended to oriented links.

How many Reidemeister moves do we need for unknotting a given diagram of the trivial knot? The absolute value of the writhe gives a lower bound of the number of Reidemeister I moves. That of a complexity of knot diagram "cowrithe" works for Reidemeister II, III moves.

In Appendix A, we give an example of an infinite sequence of diagrams D_n of the trivial knot with an O(n) number of crossings such that the author expects the number of Reidemeister moves needed for unknotting it to be O(n²). However, writhe and cowrithe do not prove this.

If a rectangular diagram represents the trivial knot, then it can be deformed into the rectangular diagram with only two vertical edges by a finite sequence of merge operations and exchange operations, without increasing the number of vertical edges, which was shown by I. A. Dynnikov. We show in this paper that we need no merge operations to deform a rectangular diagram of the trivial knot to one with no crossings.

Extending upon our previous work, we verify the Jones Unknot Conjecture for all knots up to 24 crossings. We describe the method of our approach and analyze the growth of the computational complexity of its different components.

We apply the Rasmussen spectral sequence to prove that the ${\mathbb{Z}}^3$-graded vector space structure of the HOMFLYPT homology over ${\mathbb{Z}}_2$ detects unlinks. Our proof relies on a theorem of Batson and Seed stating that the ${\mathbb{Z}}^2$-graded vector space structure of the Khovanov homology over ${\mathbb{Z}}_2$ detects unlinks.