Narrow Results

One of the spectacular results in mathematical physics is the expression of Racah matrices for symmetric representations of the quantum group ${\mathrm{\text{SU}}}_q(2)$ through the Askey–Wilson polynomials, associated with the $q$-hypergeometric functions ${}_4{\varphi }_3$. Recently it was shown that this is in fact the general property of symmetric representations, valid for arbitrary ${\mathrm{\text{SU}}}_q(N)$ — at least for exclusive Racah matrices $\bar{S}$. The natural question then is what substitutes the conventional $q$-hypergeometric polynomials when representations are more general? New advances in the theory of matrices $\bar{S}$, provided by the study of differential expansions of knot polynomials, suggest that these are multiple sums over Young sub-diagrams of the one which describes the original representation of ${\mathrm{\text{SU}}}_q(N)$. A less trivial fact is that the entries of the sum are not just the factorized combinations of quantum dimensions, as in the ordinary hypergeometric series, but involve non-factorized quantities, like the skew characters and their further generalizations — as well as associated additional summations with the Littlewood–Richardson weights.

Superconducting cosmic strings (SCSs) have received revived interests recently. In this paper we treat closed SCSs as oriented knotted line defects, and concentrate on their topology by studying the Hopf topological invariant. This invariant is an Abelian Chern–Simons action, from which the HOMFLYPT knot polynomial can be constructed. It is shown that the two independent parameters of the polynomial correspond to the writhe and twist contributions, separately. This new method is topologically stronger than the traditional (self-)linking number method, which fails to detect essential topology of knots sometimes, shedding new light upon the study of physical intercommunications of superconducting cosmic strings as a complex system.

We extend the Kauffman state models of the Jones and Alexander polynomials of classical links to state models of their two-variable extensions in the case of singular links. Moreover, we extend both of them to polynomials with d + 1 variables for long singular knots with exactly d double points. These extensions can detect non-invertibility of long singular knots.

A Gauss diagram is a simple, combinatorial way to present a link. It is known that any Vassiliev invariant may be obtained from a Gauss diagram formula that involves counting subdiagrams of certain combinatorial types. In this paper we present simple formulas for an infinite family of invariants in terms of counting surfaces of a certain genus and number of boundary components in a Gauss diagram associated with a closed braid. We then identify the resulting invariants with partial derivatives of the HOMFLY-PT polynomial.

We introduce the notion of a “state function” for framed tangles in a disk. After choosing a finite set of states for each marked disk, a state function is a projection from the vector space spanned by all tangles to the vector space spanned by the states, that is local, and topologically invariant. Given the states for the Kauffman bracket, and the quantum $SU(3)$-invariant we classify all state functions, and then compare our results to the literature.

In this paper, we examine the properties of the Jones polynomial using dimensionality reduction learning techniques combined with ideas from topological data analysis. Our data set consists of more than 10 million knots up to 17 crossings and two other special families up to 2001 crossings. We introduce and describe a method for using filtrations to analyze infinite data sets where representative sampling is impossible or impractical, an essential requirement for working with knots and the data from knot invariants. In particular, this method provides a new approach for analyzing knot invariants using Principal Component Analysis. Using this approach on the Jones polynomial data, we find that it can be viewed as an approximately three-dimensional subspace, that this description is surprisingly stable with respect to the filtration by the crossing number, and that the results suggest further structures to be examined and understood.

A special class of braids, called woven, is introduced and it is shown that every conjugation class of the braid group contains woven braids. In consequence tame links can be presented as plats and closures of woven braids. Restricting on knots we get the 'woven version' of the well-known theorem of Markov, giving moves that are capable of producing all woven braids with equivalent closures.

As corollary we obtain that a link in which each component is dyed with at least two different colors can be projected on a plane without crossing strands of the same color. The lowest order part of the HOMFLY polynomial of a closed woven braid can be read off directly; a complete characterization of all occuring terms is given.

Finally, a table of all minimal woven braids for the 84 prime knots with at most nine crossings is appended. The average word length and the average number of entries per knot type tune out to be suprising small.