Related papers: Generalized Skein Modules of Surfaces
We generalize Kauffman's famous formula defining the Jones polynomial of an oriented link in 3-space from his bracket and the writhe of an oriented diagram. Our generalization is an epimorphism between skein modules of tangles in compact…
We mainly focus on Classical limit, Splitting map, and Frobenius homomorphism for stated $SL(n)$-skein modules, and Unicity Theorem for stated $SL(n)$-skein algebras. Let $(M,N)$ be a marked three manifold. We use $S_n(M,N,v)$ to denote the…
In this paper, we introduce and study the notion of linkage of modules by reflexive homomorphisms. This notion unifies and generalizes several known concepts of linkage of modules and enables us to study the theory of linkage of modules…
Let $R$ be a commutative ring with identity and $G$ a graph. An extending generalized spline on $G$ is a vertex labeling $f \in \prod_{v} M_v$, where for each edge $e=uv$ there exists an $R$-module $M_{uv}$ together with homomorphisms $…
We explain how rank two Frobenius extensions of commutative rings lead to link homology theories and discuss relations between these theories, Bar-Natan theories, equivariant cohomology and the Rasmussen invariant.
Skein modules are the main objects of an algebraic topology based on knots (or position). In the same spirit as Leibniz we would call our approach "algebra situs." When looking at the panorama of skein modules we see, past the rolling hills…
We establish relations between Frobenius parts and between flat-dominant dimensions of algebras linked by Frobenius bimodules. This is motivated by the Nakayama conjecture and an approach of Martinez-Villa to the Auslander-Reiten conjecture…
Khovanov defined graded homology groups for links L in R^3 and showed that their polynomial Euler characteristic is the Jones polynomial of L. Khovanov's construction does not extend in a straightforward way to links in I-bundles M over…
These are expository lecture notes from a graduate topics course taught by the author on Khovanov homology and related invariants. Major topics include the Jones polynomial, Khovanov homology, Bar-Natan's cobordism category, applications of…
We construct a quantum Frobenius map for the $SL_3$ skein module of any oriented 3-manifold specialized at a root of unity, and describe the map by way of threading certain polynomials along links. The homomorphism is a higher rank version…
Link homology theories (such as knot Floer homology and Khovanov homology) have become indispensable tools for studying knots and links, including powerful 4-dimensional obstructions. These notes, based on lectures given at the 2024 Georgia…
This paper establishes an isomorphism between the Bar-Natan skein module of the solid torus with a particular boundary curve system and the homology of the (n,n) Springer variety. The results build on Khovanov's work with crossingless…
We prove a generalization of the cobordism hypothesis of Baez--Dolan and Hopkins--Lurie for bordisms with arbitrary geometric structures, such as Riemannian metrics, complex and symplectic structures, principal bundles with connections, or…
In this paper we study cobordism categories consisting of manifolds which are endowed with geometric structure. Examples of such geometric structures include symplectic structures, flat connections on principal bundles, and complex…
Khovanov homology, an invariant of links in $\mathbb{R}^3$, is a graded homology theory that categorifies the Jones polynomial in the sense that the graded Euler characteristic of the homology is the Jones polynomial. Asaeda, Przytycki and…
In this paper, we discuss degree 0 crossing change on Khovanov homology in terms of cobordisms. Namely, using Bar-Natan's formalism of Khovanov homology, we introduce a sum of cobordisms that yields a morphism on complexes of two diagrams…
Over the past thirty-seven years, the study of linear and quadratic skein modules has produced a rich and far-reaching skein theory, intricately connected to diverse areas of mathematics and physics, including algebraic geometry, hyperbolic…
We show the Frobenius pullback of a general semi-stable vector bundle in the moduli space of vector bundles with fixed rank and degree is still semi-stable by deformation trick. We then present several applications of the main theorem.
We show that relations in Homflypt type skein theory of an oriented $3$-manifold $M$ are induced from a $2$-groupoid defined from the fundamental $2$-groupoid of a space of singular links in $M$. The module relations are defined by…
We study the behaviour of the Kauffman bracket skein modules of 3-manifolds under gluing along surfaces. For this purpose we extend the notion of Kauffman bracket skein modules to $3$-manifolds with marking consisting of open intervals and…