HOMFLY-PT skein module of singular links in the three-sphere

For a ring $R$, we denote by $R[\mathcal L]$ the free $R$-module spanned by the isotopy classes of singular links in $\mathbb S^3$. Given two invertible elements $x,t \in R$, the HOMFLY-PT skein module of singular links in $\mathbb S^3$ (relative to the triple $(R,t,x)$) is the quotient of $R[\mathcal L]$ by local relations, called skein relations, that involve $t$ and $x$. We compute the HOMFLY-PT skein module of singular links for any $R$ such that $(t^{-1}-t+x)$ and $(t^{-1}-t-x)$ are invertible. In particular, we deduce the Conway skein module of singular links

On Link Homology Theories from Extended Cobordisms

This paper is devoted to the study of algebraic structures leading to link homology theories. The originally used structures of Frobenius algebra and/or TQFT are modified in two directions. First, we refine 2-dimensional cobordisms by taking into account their embedding into the three space. Secondly, we extend the underlying cobordism category to a 2-category, where the usual relations hold up to 2-isomorphisms. The corresponding abelian 2-functor is called an extended quantum field theory (EQFT). We show that the Khovanov homology, the nested Khovanov homology, extracted by Stroppel and Webster from Seidel-Smith construction, and the odd Khovanov homology fit into this setting. Moreover, we prove that any EQFT based on a Z_2-extension of the embedded cobordism category which coincides with Khovanov after reducing the coefficients modulo 2, gives rise to a link invariant homology theory isomorphic to those of Khovanov.Comment: Lots of figure

A cubic defining algebra for the Links-Gould polynomial

We define a finite-dimensional cubic quotient of the group algebra of the braid group, endowed with a (essentially unique) Markov trace which affords the Links-Grould invariant of knots and links. We investigate several of its properties, and state several conjectures about its structure

Extensions of some classical local moves on knot diagrams

In the present paper, we consider local moves on classical and welded diagrams: (self-)crossing change, (self-)virtualization, virtual conjugation, Delta, fused, band-pass and welded band-pass moves. Interrelationship between these moves is discussed and, for each of these move, we provide an algebraic classification. We address the question of relevant welded extensions for classical moves in the sense that the classical quotient of classical object embeds into the welded quotient of welded objects. As a by-product, we obtain that all of the above local moves are unknotting operations for welded (long) knots. We also mention some topological interpretations for these combinatorial quotients.Comment: 18 pages; this paper is an entirely new version of "On forbidden moves and the Delta move": the exposition has been totally revised, and several new results have been added; to appear in Michigan Math.

Homotopy classification of ribbon tubes and welded string links

Ribbon 2-knotted objects are locally flat embeddings of surfaces in 4-space which bound immersed 3-manifolds with only ribbon singularities. They appear as topological realizations of welded knotted objects, which is a natural quotient of virtual knot theory. In this paper we consider ribbon tubes and ribbon torus-links, which are natural analogues of string links and links, respectively. We show how ribbon tubes naturally act on the reduced free group, and how this action classifies ribbon tubes up to link-homotopy, that is when allowing each component to cross itself. At the combinatorial level, this provides a classification of welded string links up to self-virtualization. This generalizes a result of Habegger and Lin on usual string links, and the above-mentioned action on the reduced free group can be refined to a general "virtual extension" of Milnor invariants. As an application, we obtain a classification of ribbon torus-links up to link-homotopy.Comment: 33p. ; v2: typos and minor corrections ; v3: Introduction rewritten, exposition revised, references added. Section 5 of the previous version was significantly expanded and was separated into another paper (arXiv:1507.00202) ; v4: typos and minor corrections ; to appear in Annali della scuola Normale Superiore de Pisa (classe de scienze

On Usual, Virtual and Welded knotted objects up to homotopy

We consider several classes of knotted objects, namely usual, virtual and welded pure braids and string links, and two equivalence relations on those objects, induced by either self-crossing changes or self-virtualizations. We provide a number of results which point out the differences between these various notions. The proofs are mainly based on the techniques of Gauss diagram formulae.Comment: 14 pages. This paper is an expanded version of a former section, now removed (section 5 in versions 1 and 2) of arXiv:1407.0184. To appear in Journal of the Mathematical Society of Japa