Related papers: Double twist knots and lattice paths
A polynomial knot in $\mathbb{R}^n$ is a smooth embedding of $\mathbb{R}$ in $\mathbb{R}^n$ such that the component functions are real polynomials. In the earlier paper with Mishra, we have studied the space $\mathcal{P}$ of polynomial…
We study Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The folded ribbonlength is the length to width ratio of such a ribbon knot. We give upper bounds on the folded ribbonlength of…
We study the AJ conjecture that relates the A-polynomial and the colored Jones polynomial of a knot in $S^3$. We confirm the AJ conjecture for $(r,2)$-cables of the $m$-twist knot, for all odd integers $r$ satisfying $\begin{cases}…
For a given knot $K$ and $w>0$, we construct infinitely many mutually distinct hyperbolic knots $\{K_i\}$ such that the $P$-satellites of $K$ and $K_i$ have the same HOMFLY polynomial up to given $z$-degrees, for all braided patterns $P$…
The set of isotopy classes of nontrivial torus knots $T(p,q)$ in $S^3$ is in bijection with the set of coprime integer pairs $(p,q)$ satisfying $|p|>q\geq 2$. We verify the AJ conjecture for the connected sums $T(p,q)\# T(a,b)$ when $p$ and…
Recently, a plethora of multivariable knot polynomials were introduced by Kashaev and one of the authors, by applying the Reshetikhin-Turaev functor to rigid $R$-matrices that come from braided Hopf algebras with automorphisms. We study the…
We introduce the deformed fermionic numbers, corresponding to the skein relations, the main characteristics of knots and links. These fermionic numbers allow one to restore the skein relations. For the Alexander (Jones) skein relation we…
Suppose $G$ is a simple graph with $n$ vertices, $m$ edges, and rank $r$. Let $\chi_G(t)=a_0t^n-a_1t^{n-1}+\cdots +(-1)^ra_rt^{n-r}$ be the chromatic polynomial of $G$. For $q,k\in \Bbb{Z}$ and $0\le k\le q+r+1$, we obtain a sharp two-side…
We investigate possibilities of generalizing the TBEM eigenvalue matrix model, which represents the non-normalized colored HOMFLY polynomials for torus knots as averages of the corresponding characters. We look for a model of the same type,…
We introduce topological invariants of knots and braid conjugacy classes, in the form of differential graded algebras, and present an explicit combinatorial formulation for these invariants. The algebras conjecturally give the relative…
Quantum knot invariants (like colored HOMFLY-PT or Kauffman polynomials) are a distinguished class of non-perturbative topological invariants. Any known way to construct them (via Chern-Simons theory or quantum R-matrix) starts with a…
We introduce and compute a 2-parameter family deformation of the A-polynomial that encodes the color dependence of the superpolynomial and that, in suitable limits, reduces to various deformations of the A-polynomial studied in the…
We present a comprehensive classification of invariants of knots and links associated with irreducible representations of \uqslN{}, when the parameter of quantization $q$ is a root of unity. We demonstrate that, besides the standard…
A classical knot is described by a one-stroke trajectory with entanglements of a string. The replica method appears as a powerful tool in statistical mechanics for a polymer or self-avoiding walk. We consider this replica N to 0 limit in…
We prove that the generating functions for the one row/column colored HOMFLY-PT invariants of arborescent links are specializations of the generating functions of the motivic Donaldson-Thomas invariants of appropriate quivers that we…
We sketch a construction of Legendrian Symplectic Field Theory (SFT) for conormal tori of knots and links. Using large $N$ duality and Witten's connection between open Gromov-Witten invariants and Chern-Simons gauge theory, we relate the…
New bispectral polynomials orthogonal on a Bannai-Ito bi-lattice (uniform quadri-lattice) are obtained from an unconventional truncation of the untruncated Bannai-Ito and complementary Bannai-Ito polynomials. A complete characterization of…
In this paper, we describe properties of the characteristic polynomial of a weighted lattice and show that it has a recursive description, which we use to obtain results on the critical exponent of $q$-polymatroids. We give a Critical…
We construct new knot polynomials. Let $V$ be the standard solid torus in 3-space and let $pr$ be its standard projection onto an annulus. Let $M$ be the space of all smooth oriented knots in $V$ such that the restriction of $pr$ is an…
We consider a class of lattice paths with certain restrictions on their ascents and down steps and use them as building blocks to construct various families of Dyck paths. We let every building block $P_j$ take on $c_j$ colors and count all…