Related papers: The bracket polynomial of bonded knots and applica…
We equip a knot $K$ with a set of colored bonds, that is, colored intervals properly embedded into $\mathbb{R}^3 \setminus K$. Such a construction can be viewed as a structure that topologically models a closed protein chain including any…
We present a systematic classification of uncolored bonded knots with singularity number at most seven. Bonded knots provide a topological model for closed protein chains with intramolecular bridges, such as disulfide bonds. Following 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…
The Kauffman bracket skein module $K(M)$ of a 3-manifold $M$ is defined over formal power series in the variable $h$ by letting $A=e^{h/4}$. For a compact oriented surface $F$, it is shown that $K(F \times I)$ is a quantization of the…
We define two new invariants for tied links. One of them can be thought as an extension of the Kauffman polynomial and the other one as an extension of the Jones polynomial which is constructed via a bracket polynomial for tied links. These…
Kauffman's bracket is an invariant of regular isotopy of knots and links which since its discovery in 1985 it has been used in many different directions: (a) it implies an easy proof of the invariance of (in fact, it is equivalent to) the…
In this paper we present recent results on the computation of skein modules of 3-manifolds using braids and appropriate knot algebras. Skein modules generalize knot polynomials in $S^3$ to knot polynomials in arbitrary 3-manifolds and they…
Proteins are linear molecular chains that often fold to function. The topology of folding is widely believed to define its properties and function, and knot theory has been applied to study protein structure and its implications. More that…
Bonded knots arise naturally in topological protein modeling, where intramolecular interactions such as disulfide bridges stabilize folded configurations. These structures extend classical knot theory by incorporating embedded graphs, and…
In this paper we present a detailed study of \emph{bonded knots} and their related structures, integrating recent developments into a single framework. Bonded knots are classical knots endowed with embedded bonding arcs modeling physical or…
We compute the Kauffman bracket skein module of the complement of a twist knot, finding that it is free and infinite dimensional. The basis consists of cables of a two-component link, one component of which is a meridian of the knot. The…
This work presents formulas for the Kauffman bracket and Jones polynomials of 3-bridge knots using the structure of Chebyshev knots and their billiard table diagrams. In particular, these give far fewer terms than in the Skein relation…
In this paper we present two different ways for computing the Kauffman bracket skein module of $S^1\times S^2$, ${\rm KBSM}\left(S^1\times S^2\right)$, via braids. We first extend the universal Kauffman bracket type invariant $V$ for knots…
We generalize Kauffman's famous formula defining the Jones polynomial of an oriented link in 3-space from his bracket and the writhe of an oriented diagram. Our generalization is an epimorphism between skein modules of tangles in compact…
The classical Thistlethwaite theorem for links can be phrased as asserting that the Kauffman bracket of a link can be obtained from an evaluation of the Bollob\'as-Riordan polynomial of a ribbon graph associated to one of the link's…
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…
Given any unoriented link diagram, a group of new knot invariants are constructed. Each of them satisfies a generalized 4 term skein relation. The coefficients of each invariant is from a commutative ring. Homomorphisms and representations…
The Kauffman bracket skein module $K(M)$ of a $3$-manifold $M$ is the quotient of the $\mathbb{Q}(A)$-vector space spanned by isotopy classes of links in $M$ by the Kauffman relations. A conjecture of Witten states that if $M$ is closed…
In this paper, we formulate a construction of ideal coset invariants for surface-links in $4$-space using invariants for knots and links in $3$-space. We apply the construction to the Kauffman bracket polynomial invariant and obtain an…
We introduce a class of links whose bracket polynomials admit an expansion over perfect matchings of a plane bipartite graph. This class includes 2-bridge links, pretzel links, and Montesinos links. Our first main result (Theorem A)…