We will show that deformation quantization in skein theory of oriented 3-manifolds is induced from a topological deformation quantization of the fundamental 2-groupoid of the space of immersions of circles in M . The structure of skein module and its relations with string topology homomorphisms appear through representations of the groupoid structure into the set the objects. The deformation of...

Algebra Situs is a branch of mathematics which has its roots in Jones’ construction of his polynomial invariant of links and Drinfeld’s work on quantum groups. It encompasses the theory of quantum invariants of knots and 3-manifolds, algebraic topology based on knots, operads, planar algebras, q-deformations, quantum groups, and overlaps with algebraic geometry, non-commutative geometry and sta...

We present the new skein invariants of classical links, H[H], K[K] and D[D], based on the invariants of links, H, K and D, denoting the regular isotopy version of the Homflypt polynomial, the Kauffman polynomial and the Dubrovnik polynomial. The invariants are obtained by abstracting the skein relation of the corresponding invariant and making a new skein algorithm comprising two computational ...

The hash function Skein is one of 5 finalists of the NIST SHA-3 competition. It is based on the block cipher Threefish which only uses three primitive operations: modular addition, rotation and bitwise XOR (ARX). This paper proposes a free-start partial-collision attack on round-reduced Skein-256 by combing the rebound attack with the modular differential techniques. The main idea of our attack...

Abstract. Let k be a subring of the field of rational functions in α, s which contains α, s. Let M be a compact oriented 3-manifold, and let K(M) denote the Kauffman skein module of M over k. Then K(M) is the free k-module generated by isotopy classes of framed links in M modulo the Kauffman skein relations. In the case of k = Q(α, s), the field of rational functions in α, s, we give a basis fo...

The hash function Skein is the submission of Ferguson et al. to the NIST Hash Competition, and is arguably a serious candidate for selection as SHA-3. This paper presents the first third-party analysis of Skein, with an extensive study of its main component: the block cipher Threefish. We notably investigate near collisions, distinguishers, impossible differentials, key recovery using related-k...

Let k be an integral domain containing the invertible elements α, s and 1 s−s −1 . If M is an oriented 3-manifold, let K(M) denote the Kauffman skein module of M over k. Based on the work on Birman-Murakami-Wenzl algebra by Beliakova and Blanchet [2], we give an “idempotent-like” basis for the Kauffman skein module of handlebodies. Gilmer and Zhong [6] have studied the Homflypt skein modules of...

Abstract. If M is a compact oriented 3-manifold, let S(M) denote the Homflypt skein module of M. We show that S(M1#M2) is isomorphic to S(M1)⊗ S(M2) modulo torsion. In fact, we show that S(M1#M2) is isomorphic to S(M1) ⊗ S(M2) if we are working over a certain localized ring. We show a similar result holds for relative skein modules. If M contains a separating 2-sphere, we give conditions under ...

The theory of link-homotopy, introduced by Milnor, is an important part of the knot theory, with Milnor’s μ̄-invariants being the basic set of link-homotopy invariants. Skein relations for knot and link invariants played a crucial role in the recent developments of knot theory. However, while skein relations for Alexander and Jones invariants are known for quite a while, a similar treatment of M...

We give an explicit correspondence between stated skein algebras, which are defined via relations on tangles in [Costantino F., L\^e T.T.Q., arXiv:1907.11400], and internal as endomorphism algebras free cocompletions of categories [Ben-Zvi D., Brochier A., Jordan J. Topol. 11 (2018), 874-917, arXiv:1501.04652] or [Gunningham S., Safronov P., arXiv:1908.05233]. Stated surfaces with multiple boun...