# Categorification: tangle invariants and TQFTs

@inproceedings{Stroppel2022CategorificationTI, title={Categorification: tangle invariants and TQFTs}, author={Catharina Stroppel}, year={2022} }

Based on diﬀerent views on the Jones polynomial we review representation theoretic categoriﬁed link and tangle invariants. We unify them in a common combinatorial framework and connect them via the theory of Soergel bimodules. The inﬂuence of these categoriﬁcations on the development of 2-representation theory and the interaction between topological invariants and 2-categorical structures is discussed. Finally, we indicate how categoriﬁed representations of quantum groups on the one hand and…

## One Citation

### Howe duality and dynamical Weyl group

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## References

SHOWING 1-10 OF 138 REFERENCES

### On the functoriality of sl(2) tangle homology.

- Mathematics
- 2019

We construct an explicit equivalence between the (bi)category of gl(2) webs and foams and the Bar-Natan (bi)category of Temperley-Lieb diagrams and cobordisms. With this equivalence we can fix…

### A functor-valued invariant of tangles

- Mathematics
- 2002

We construct a family of rings. To a plane diagram of a tangle we associate a complex of bimodules over these rings. Chain homotopy equivalence class of this complex is an invariant of the tangle. On…

### Category O and slk link invariants

- Mathematics
- 2007

We construct a functor valued invariant of oriented tangles on certain singular blocks of category O. Parabolic subcategories of these blocks categorify tensor products of various fundamental sl(k)…

### Khovanov's homology for tangles and cobordisms

- Mathematics
- 2004

We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2-knots. By staying within a world of topological…

### Categorified Young symmetrizers and stable homology of torus links

- MathematicsGeometry & Topology
- 2018

We construct complexes $P_{1^n}$ of Soergel bimodules which categorify the Young idempotents corresponding to one-column partitions. A beautiful recent conjecture of Gorsky-Rasmussen relates the…

### Categorification of the Temperley-Lieb category, tangles, and cobordisms via projective functors

- Mathematics
- 2005

To each generic tangle projection from the three-dimensional real vector space onto the plane, we associate a derived endofunctor on a graded parabolic version of the Bernstein-Gel'fand category…

### Triply-graded link homology and Hochschild homology of Soergel bimodules

- Mathematics
- 2005

We consider a class of bimodules over polynomial algebras which were originally introduced by Soergel in relation to the Kazhdan–Lusztig theory, and which describe a direct summand of the category of…

### Tilting modules and the p-canonical basis

- Mathematics
- 2015

In this paper we propose a new approach to tilting modules for reductive algebraic groups in positive characteristic. We conjecture that translation functors give an action of the (diagrammatic)…

### Matrix factorizations and link homology

- Mathematics
- 2008

Author(s): Khovanov, Mikhail; Rozansky, Lev | Abstract: For each positive integer n the HOMFLY polynomial of links specializes to a one-variable polynomial that can be recovered from the…

### A categorification of finite-dimensional irreducible representations of quantum sl(2) and their tensor products

- Mathematics
- 2005

The purpose of this paper is to study categorifications of tensor products of finite dimensional modules for the quantum group for sl(2). The main categorification is obtained using certain…