An invariant of tangle cobordisms
@article{Khovanov2002AnIO, title={An invariant of tangle cobordisms}, author={Mikhail Khovanov}, journal={Transactions of the American Mathematical Society}, year={2002}, volume={358}, pages={315-327} }
We construct a new invariant of tangle cobordisms. The invariant of a tangle is a complex of bimodules over certain rings, well-defined up to chain homotopy equivalence. The invariant of a tangle cobordism is a homomorphism between complexes of bimodules assigned to boundaries of the' cobordism.
14 Citations
An invariant of tangle cobordisms via subquotients of arc rings
- Mathematics
- 2006
We construct an explicit categorification of the action of tangles on tensor powers of the fundamental representation of quantum sl(2).
A Burau-Alexander 2-functor on tangles
- Mathematics
- 2016
We construct a weak 2-functor from the bicategory of oriented tangles to a bicategory of Lagrangian cospans. This functor simultaneously extends the Burau representation of the braid groups, its…
Functoriality of Khovanov homology
- Mathematics
- 2015
In this paper we prove that every Khovanov homology associated to a Frobenius algebra of rank $2$ can be modified in such a way as to produce a TQFT on oriented links, that is a monoidal functor from…
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…
A Lie theoretic categorification of the coloured Jones polynomial
- MathematicsJournal of Pure and Applied Algebra
- 2022
An oriented model for Khovanov homology
- Mathematics
- 2010
We give an alternative presentation of Khovanov homology of links with strict functoriality result over integers. The construction uses an oriented $sl(2)$ state model allowing a natural definition…
An sl(2) tangle homology and seamed cobordisms
- Mathematics
- 2007
We construct a bigraded (co)homology theory which depends on a parameter a, and whose graded Euler characteristic is the quantum sl(2) link invariant. We follow Bar-Natan's approach to tangles on one…
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…
sl(2) tangle homology with a parameter and singular cobordisms
- Mathematics
- 2008
We construct a bigraded cohomology theory which depends on one parameter a, and whose graded Euler characteristic is the quantum sl.2/ link invariant. We follow Bar-Natan’s approach to tangles on one…
References
SHOWING 1-10 OF 13 REFERENCES
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…
CROSSINGLESS MATCHINGS AND THE COHOMOLOGY OF (n, n) SPRINGER VARIETIES
- Mathematics
- 2002
In an earlier paper, the author introduced a collection of rings that control a categorification of the quantum sl(2) invariant of tangles. We prove that centers of these rings are isomorphics to the…
An invariant of link cobordisms from Khovanov's homology theory
- Mathematics
- 2002
Mikhail Khovanov constructed (math.QA/9908171) a homology theory of oriented links, which has the Jones polynomial as its graded Euler characteristic. He also explained how every link cobordism…
Reidemeister-type moves for surfaces in four-dimensional space
- Mathematics
- 1998
We consider smooth knottings of compact (not necessarily orientable) n-dimensional manifolds in R (or S), for the cases n = 2 or n = 3. In a previous paper we have generalized the notion of the…
A Combinatorial Description of Knotted Surfaces and Their Isotopies
- Mathematics
- 1997
Abstract We discuss the diagrammatic theory of knot isotopies in dimension 4. We project a knotted surface to a three-dimensional space and arrange the surface to have generic singularities upon…
Knotted Surfaces and Their Diagrams
- Mathematics
- 1997
Diagrams of knotted surfaces Moving knotted surfaces Braid theory in dimension four Combinatorics of knotted surface diagrams The fundamental group and the Seifert algorithm Algebraic structures…
REIDEMEISTER MOVES FOR SURFACE ISOTOPIES AND THEIR INTERPRETATION AS MOVES TO MOVIES
- Mathematics
- 1993
A movie description of a surface embedded in 4-space is a sequence of knot and link diagrams obtained from a projection of the surface to 3-space by taking 2-dimensional cross sections perpendicular…
On the Definition of $2$-Category of $2$-Knots
- Mathematics
- 1993
© Université Louis Pasteur (Strasbourg), 1993, tous droits réservés. L’accès aux archives de la série « Recherche Coopérative sur Programme no 25 » implique l’accord avec les conditions générales…