Related papers: Explicit formulas for the Vassiliev knot invariant…
Pulling back the weight systems associated with the exceptional Lie algebras and their standard representations by a modification of the universal Vassiliev-Kontsevich invariant yields link invariants; extending them to coloured 3-nets, we…
We introduce a new invariant for a $2$-knot in $S^4$, called the shadow-complexity, based on the theory of Turaev shadows, and we give a characterization of $2$-knots with shadow-complexity at most $1$. Specifically, we show that the unknot…
In this paper we define and give examples of a family of polynomial invariants of virtual knots and links. They arise by considering certain 2$\times$2 matrices with entries in a possibly non-commutative ring, for example the quaternions.…
We find that Koschorke's $\beta$-invariant and the triple $\mu$-invariant of link maps in the critical dimension can be computed as degrees of certain maps of configuration spaces - just like the linking number. Both formulas admit…
The Turaev-Viro invariants are a powerful family of topological invariants for distinguishing between different 3-manifolds. They are invaluable for mathematical software, but current algorithms to compute them require exponential time. The…
We obtain a localized version of the configuration space integral for the Casson knot invariant, where the standard symmetric Gauss form is replaced with a locally supported form. An interesting technical difference between the arguments…
We show that Vassiliev invariants separate braids on a closed oriented surface, and we exhibit an universal Vassiliev invariant for these braids in terms of chord diagrams labeled by elements of the fundamental group of the considered…
We introduce a new combinatorial method to encode knots and links with applications to knot invariants. Clasp diagrams defined in this paper are combinatorial blueprints for building knot diagrams out of full twists on two strings rather…
A brief summary of the development of perturbative Chern-Simons gauge theory related to the theory of knots and links is presented. Emphasis is made on the progress achieved towards the determination of a general combinatorial expression…
The Kontsevich integral of a knot is a powerful invariant which takes values in an algebra of trivalent graphs with legs. Given a Lie algebra, the Kontsevich integral determines an invariant of knots (the so-called colored Jones function)…
We provide a geometric construction of the boundary states for handlebodies which we in turn use to give a geometric formula for the Witten-Reshetikhin-Turaev quantum invariants. We then analyze the asymptotics of this invariant in the…
We consider a special class of Kauffman's graph invariants of rigid vertex isotopy (graph invariants of Vassiliev type). They are given by a functor from a category of colored and oriented graphs embedded into a 3-space to a category of…
We prove a refinement of Vogel's statement that the Vassiliev invariants of knots coming from semisimple Lie algebras do not generate all Vassiliev invariants. This refinement takes into account the second grading on Vassiliev invariants…
In this paper we prove that the $r$-th ADO polynomial of a knot, for $r$ a power of prime number, can be expanded as Vassiliev invariants with values in $\mathbb{Z}$. Nevertheless this expansion is not unique and not easily computable. We…
The total twist number, which represents the first non-trivial Vassiliev knot invariant, is derived from the second order expression of the Wilson loop expectation value in the Chern-Simons theory. Using the well-known fact that the…
We prove that braid invariants coming from quantum gl(N) separate braids, by recalling that these invariants (properly decomposed) are all Vassiliev invariants, showing that all Vassiliev invariants of braids arise in this way, and…
We study the unwheeled rational Kontsevich integral of torus knots. We give a precise formula for these invariants up to loop degree 3 and show that they appear as colorings of simple diagrams. We show that they behave under cyclic branched…
We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl(2) and sl(3) and by…
We give a general fixed parameter tractable algorithm to compute quantum invariants of links presented by diagrams, whose complexity is singly exponential in the carving-width (or the tree-width) of the diagram. In particular, we get a…
We look into computational aspects of two classical knot invariants. We look for ways of simplifying the computation of the coloring invariant and of the Alexander module. We support our ideas with explicit computations on pretzel knots.