Narrow Results

We numerically estimate the superposition of the HOMFLY-PT polynomial of an open two-component link, define its spread, and describe how this quantity may be employed to quantify the degree of entanglement of confined two component open links.

This text is a brief description of the Jones polynomial as it appears in the announcement in [9], together with Jones’ comments in the letter he addressed to Joan Birman on 31st May 1984 [8]. As much as possible I have followed the author’s initial notations. I add comments about the history of the events as I remember them, with the help of several testimonies of the “actors”. Among them: Didier Hatt-Arnold, Pierre de la Harpe, Cam Van Quach Hongler. It took me nontrivial efforts with my memories to get the chronology right (I hope so). The help of Eva Bayer was decisive. My warmest thanks to her. One may disagree with some of my statements which express a personal opinion. On purpose I have adopted a candid style.

We study the existence of relations between the degrees of the knot polynomials and some classical knot invariants, partially confirming and extending the question of Morton on the skein polynomial and a recent question of Ferrand.

Cosmic strings in the early universe have received revived interest in recent years. In this paper, we derive these structures as topological defects from singular distributions of the quintessence field of dark energy. Our emphasis is placed on the topological charge of tangled cosmic strings, which originates from the Hopf mapping and is a Chern–Simons action possessing strong inherent tie to knot topology. It is shown that the Kauffman bracket knot polynomial can be constructed in terms of this charge for unoriented knotted strings, serving as a topological invariant much stronger than the traditional Gauss linking numbers in characterizing string topology. Especially, we introduce a mathematical approach of breaking-reconnection which provides a promising candidate for studying physical reconnection processes within the complexity-reducing cascades of tangled cosmic strings.

In 2006 C. Hayashi gave a lower bound for the number of Reidemeister moves in deformation of two equivalent knot diagrams by using writhe and cowrithe. It can be naturally extended for two virtually isotopic virtual knot diagrams. We introduce a polynomial q_K(t) of a virtual knot diagram K and give lower bounds for the number of Reidemeister moves in deformation of two virtually isotopic knots by using q_K(t). We give an example which shows that the polynomial q_K(t) is useful to map out a sequence of Reidemeister moves to deform a virtual knot diagram to another virtually isotopic one.

When two virtual knot diagrams are virtually isotopic, there is a sequence of Reidemeister moves and virtual moves relating them. I introduced a polynomial $q_K(t)$ of a virtual knot diagram $K$ and gave lower bounds for the number of Reidemeister moves in deformation of virtually isotopic knot diagrams by using $q_K(t)$. In this paper, I introduce bridge diagrams and polynomials of virtual knot diagrams based on parity of crossings, and show that the polynomials give lower bounds for the number of the third Reidemeister moves. I give an example which shows that the result is distinguished from that obtained from $q_K(t)$.

Given any oriented link diagram, one can construct knot invariants using skein relations. Usually such a skein relation contains three or four terms. In this paper, the author introduces several new ways to smooth a crossings, and uses a system of skein equations to construct link invariants. This invariant can also be modified by writhe to get a more powerful invariant. The modified invariant is a generalization of both the HOMFLYPT polynomial and the two-variable Kauffman polynomial. Using the diamond lemma, a simplified version of the modified invariant is given. It is easy to compute and is a generalization of the two-variable Kauffman polynomial.

We give a new upper bound on the maximum degree of the Jones polynomial of a fibered positive link. In particular, we prove that the maximum degree of the Jones polynomial of a fibered positive knot is at most four times the minimum degree. Using this result, we can complete the classification of all knots of crossing number $\le 12$ as positive or not positive, by showing that the seven remaining knots for which positivity was unknown are not positive. That classification was also done independently at around the same time by Stoimenow.