Related papers: Invariants of Random Knots and Links
We define relative versions of the classical invariants of Legendrian and transverse knots in contact 3-manifolds for knots that are homologous to a fixed reference knot. We show these invariants are well-defined and give some basic…
Knot and link polynomials are topological invariants calculated from the expectation value of loop operators in topological field theories. In 3D Chern-Simons theory, these invariants can be found from crossing and braiding matrices of…
We propose a new method for numerical calculation of link plynomials for knots given in 3 dimensions. We calculate derivatives of the Jones polynomial in a computational time proportional to $N^{\alpha}$ with respect to the system size $N$…
We introduce new skein invariants of links based on a procedure where we first apply the skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using a…
Notes based on lessons given at {\sc Escuela " Fico Gonz\'alez Acu\~na" de Nudos y 3-variedades}, M\'erida Yucat\'an, M\'exico, 7--10 (2015) and {\sc Encuentro de nudos, trenzas y \'algebras}, Oaxaca--M\'exico, 3--10 October (2018).
We introduce new topological quantum invariants of compact oriented 3-manifolds with boundary where the boundary is a disjoint union of two identical surfaces. The invariants are constructed via surgery on manifolds of the form $F \times I$…
The lectures review the state of affairs in modern branch of mathematical physics called probabilistic topology. In particular we consider the following problems: (i) We estimate the probability of a trivial knot formation on the lattice…
A Fox p-colored knot $K$ in $S^3$ gives rise to a $p$-fold branched cover $M$ of $S^3$ along $K$. The pre-image of the knot $K$ under the covering map is a $\dfrac{p+1}{2}$-component link $L$ in $M$, and the set of pairwise linking numbers…
We study petal diagrams of knots, which provide a method of describing knots in terms of permutations in a symmetric group $S_{2n+1}$. We define two classes of moves on such permutations, called trivial petal additions and crossing…
To each link $L$ in $S^3$ we associate a collection of certain labelled directed trees, called width trees. We interpret some classical and new topological link invariants in terms of these width trees and show how the geometric structure…
We study RNA foldings and investigate their topology using a combination of knot theory and embedded rigid vertex graphs. Knot theory has been helpful in modeling biomolecules, but classical knots place emphasis on a biomolecule's…
We explore free knot diagrams, which are projections of knots into the plane which don't record over/under data at crossings. We consider the combinatorial question of which free knot diagrams give which knots and with what probability.…
In this article, we introduce rack invariants of oriented Legendrian knots in the 3-dimensional Euclidean space endowed with the standard contact structure, which we call Legendrian racks. These invariants form a generalization of the…
In a previous work, the first and third authors studied a random knot model for all two-bridge knots using billiard table diagrams. Here we present a closed formula for the distribution of the crossing numbers of such random knots. We also…
2-dimensional knots and links are studied in the article. The notion of parity is introduced via techniques similar to the ones used by the second named author in 1-dimensional case. By using parity new invariants are constructed and known…
Carter, Jelsovsky, Kamada, Langford and Saito have defined an invariant of classical links associated to each element of the second cohomology of a finite quandle. We study these invariants for Alexander quandles of the form Z[t,t^{-1}]/(p,…
We initiate the study of classical knots through the homotopy class of the n-th evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its n-th evaluation map realizes the…
We introduce and study knots and links in 2-dimensional complexes. In particular, we define linking numbers for oriented two-component links in 2-complexes and a Kauffman-type bracket polynomial for links in 2-complexes. We also discuss…
We compute the invariants for a class of knots and links in arbitrary representations in $S^3/\mathbb{Z}_p$ in the large $k$ (level), large $N$ (rank) limit, keeping $N/(k+N)=\lambda$ fixed, in $U(N)$ and $Sp(N)$ Chern-Simons theories.…
Chern-Simons theories, which are topological quantum field theories, provide a field theoretic framework for the study of knots and links in three dimensions. These are rare examples of quantum field theories which can be exactly and…