Topological steps toward the Homflypt skein module of the lens spaces L(p,1)L(p,1) via braids

In this paper we work toward the Homflypt skein module of the lens spaces $L(p,1)$, $\mathcal{S}(L(p,1))$, using braids. In particular, we establish the connection between $\mathcal{S}({\rm ST})$, the Homflypt skein module of the solid torus ST, and $\mathcal{S}(L(p,1))$ and arrive at an infinite system, whose solution corresponds to the computation of $\mathcal{S}(L(p,1))$. We start from the Lambropoulou invariant $X$ for knots and links in ST, the universal analogue of the Homflypt polynomial in ST, and a new basis, $\Lambda$, of $\mathcal{S}({\rm ST})$ presented in \cite{DL1}. We show that $\mathcal{S}(L(p,1))$ is obtained from $\mathcal{S}({\rm ST})$ by considering relations coming from the performance of braid band moves (bbm) on elements in the basis $\Lambda$, where the braid band moves are performed on any moving strand of each element in $\Lambda$. We do that by proving that the system of equations obtained from diagrams in ST by performing bbm on any moving strand is equivalent to the system obtained if we only consider elements in the basic set $\Lambda$. The importance of our approach is that it can shed light to the problem of computing skein modules of arbitrary c.c.o. $3$-manifolds, since any $3$-manifold can be obtained by surgery on $S^3$ along unknotted closed curves. The main difficulty of the problem lies in selecting from the infinitum of band moves some basic ones and solving the infinite system of equations.Comment: 24 pages, 16 figures. arXiv admin note: text overlap with arXiv:1412.364

An important step for the computation of the HOMFLYPT skein module of the lens spaces L(p,1)L(p,1) via braids

We prove that, in order to derive the HOMFLYPT skein module of the lens spaces $L(p,1)$ from the HOMFLYPT skein module of the solid torus, $\mathcal{S}({\rm ST})$, it suffices to solve an infinite system of equations obtained by imposing on the Lambropoulou invariant $X$ for knots and links in the solid torus, braid band moves that are performed only on the first moving strand of elements in a set $\Lambda^{aug}$, augmenting the basis $\Lambda$ of $\mathcal{S}({\rm ST})$.Comment: 20 pages, 5 figures. arXiv admin note: substantial text overlap with arXiv:1702.06290, arXiv:1604.0616

Canonical bases and higher representation theory

This paper develops a general theory of canonical bases, and how they arise naturally in the context of categorification. As an application, we show that Lusztig's canonical basis in the whole quantized universal enveloping algebra is given by the classes of the indecomposable 1-morphisms in a categorification when the associated Lie algebra is finite type and simply laced. We also introduce natural categories whose Grothendieck groups correspond to the tensor products of lowest and highest weight integrable representations. This generalizes past work of the author's in the highest weight case.Comment: 55 pages; DVI may not compile correctly, PDF is preferred. v2: added section on dual canonical bases. v3: improved exposition in line with new version of 1309.3796. v4: final version, to appear in Compositio Mathematica. v5: corrected references for proof of Theorem 4.

On the cohomology of spaces of links and braids via configuration space integrals

We study the cohomology of spaces of string links and braids in $\mathbb{R}^n$ for $n\geq 3$ using configuration space integrals. For $n>3$, these integrals give a chain map from certain diagram complexes to the deRham algebra of differential forms on these spaces. For $n=3$, they produce all finite type invariants of string links and braids.Comment: 20 page

On the Relationship between Strand Spaces and Multi-Agent Systems

Strand spaces are a popular framework for the analysis of security protocols. Strand spaces have some similarities to a formalism used successfully to model protocols for distributed systems, namely multi-agent systems. We explore the exact relationship between these two frameworks here. It turns out that a key difference is the handling of agents, which are unspecified in strand spaces and explicit in multi-agent systems. We provide a family of translations from strand spaces to multi-agent systems parameterized by the choice of agents in the strand space. We also show that not every multi-agent system of interest can be expressed as a strand space. This reveals a lack of expressiveness in the strand-space framework that can be characterized by our translation. To highlight this lack of expressiveness, we show one simple way in which strand spaces can be extended to model more systems.Comment: A preliminary version of this paper appears in the Proceedings of the 8th ACM Conference on Computer and Communications Security,200