Related papers: Universal geometrical link invariants
We extend the Gordon-Litherland pairing to links in thickened surfaces, and use it to define signature, determinant, and nullity invariants for links that bound (unoriented) spanning surfaces. The invariants are seen to depend only on the…
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 a new approach to universal quantum knot invariants that emphasizes generating functions instead of generators and relations. All the relevant generating functions are shown to be perturbed Gaussians of the form $Pe^G$, where…
A polynomial is presented that models a topological knot in a unique manner. It distinguishes all types of knots including the orientation and has a group theory interpretation. The topologies may be labeled via a number, which upon a base…
This is a survey talk on one of the best known quantum knot invariants, the colored Jones polynomial of a knot, and its relation to the algebraic/geometric topology and hyperbolic geometry of the knot complement. We review several aspects…
The Jones polynomial is a famous link invariant that can be defined diagrammatically via a skein relation. Khovanov homology is a richer link invariant that categorifies the Jones polynomial. Using spectral sequences, we obtain a skein-type…
In this paper we give a re-normalization of the Reshetikhin-Turaev quantum invariants of links, by modified quantum dimensions. In the case of simple Lie algebras these modified quantum dimensions are proportional to the usual quantum…
An elementary introduction to knot theory and its link to quantum field theory is presented with an intention to provide details of some basic calculations in the subject, which are not easily found in texts. Study of Chern-Simons theory…
Twisted links are a generalization of classical links and correspond to stably equivalence classes of links in thickened surfaces. In this paper we introduce twisted intersection colorings of a diagram and construct two invariants of a…
A categorification of a polynomial link invariant is an homological invariant which contains the polynomial one as its graded Euler characteristic. This field has been initiated by Khovanov categorification of the Jones polynomial. Later,…
Tied links and the tied braid monoid were introduced recently by the authors and used to define new invariants for classical links. Here, we give a version purely algebraic-combinatoric of tied links. With this new version we prove that the…
We study the asymptotic expansion of the colored Jones polynomial (the Melvin-Morton expansion) using a recursion formula for the deframed universal weight system for the $sl(2)$ Lie algebra. Combined with the formula for the universal…
We show that embedding calculus invariants $ev_n$ are surjective for long knots in an arbitrary $3$-manifold. This solves some remaining open cases of Goodwillie--Klein--Weiss connectivity estimates, and at the same time confirms one half…
It has been argued based on electric-magnetic duality and other ingredients that the Jones polynomial of a knot in three dimensions can be computed by counting the solutions of certain gauge theory equations in four dimensions. Here, we…
We define a new topological invariant of line arrangements in the complex projective plane. This invariant is a root of unity defined under some combinatorial restrictions for arrangements endowed with some special torsion character on the…
We construct two complete invariants of oriented classical knots in space. The value of each invariant on any knot is a set, infinite for the first invariant and finite for the second. The finite set is computed algorithmically from any…
Several classical knot invariants, such as the Alexander polynomial, the Levine-Tristram signature and the Blanchfield pairing, admit natural extensions from knots to links, and more generally, from oriented links to so-called colored…
The knot coloring polynomial defined by Eisermann for a finite pointed group is generalized to an infinite pointed group as the longitudinal mapping invariant of a knot. In turn this can be thought of as a generalization of the quandle…
One construction of the Alexander polynomial is as a quantum invariant associated with representations of restricted quantum $\mathfrak{sl}_2$ at a fourth root of unity. We generalize this construction to define a link invariant…
We investigate the $q$-holonomic properties of a class of link invariants based on quantum group representations with vanishing quantum dimensions, motivated by the search for the invariants' realization in physics. Some of the best known…