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Register a new userAn invariant of tangle cobordisms via subquotients of arc rings
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We construct an explicit categorification of the action of tangles on tensor powers of the fundamental representation of quantum sl(2).
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We build extensions of the arc rings, relate their centers to the cohomology rings of the Springer varieties, and categorify all level two representations of quantum sl(N).
We introduce a generalization of oriented tangles, which are still called tangles, so that they are in one-to-one correspondence with the sutured manifolds. We define cobordisms between sutured manifolds (tangles) by generalizing cobordisms between oriented tangles. For every commutative algebra A over Z/2Z, we define A-Tangles to be the category consisting of A-tangles, which are balanced tangles with A-colorings of the tangle strands and fixed SpinC structures, and A-cobordisms as morphisms. An A-cobordism is a cobordism with a compatible A-coloring and an affine set of SpinC structures. Associated with every A-module M we construct a functor $HF^M$ from A-Tangles to A-Modules, called the tangle Floer homology functor, where A-Modules denotes the the category of A-modules and A-homomorphisms between them. Moreover, for any A-tangle T the A-module $HF^M(T)$ is the extension of sutured Floer homology defined in an earlier work of the authors. In particular, this construction generalizes the 4-manifold invariants of Ozsvath and Szabo. Moreover, applying the above machinery to decorated cobordisms between links, we get functorial maps on link Floer homology.
Lagrangian cobordisms between Legendrian knots arise in Symplectic Field Theory and impose an interesting and not well-understood relation on Legendrian knots. There are some known elementary building blocks for Lagrangian cobordisms that are smoothly the attachment of $0$- and $1$-handles. An important question is whether every pair of non-empty Legendrians that are related by a connected Lagrangian cobordism can be related by a ribbon Lagrangian cobordism, in particular one that is decomposable into a composition of these elementary building blocks. We will describe these and other combinatorial building blocks as well as some geometric methods, involving the theory of satellites, to construct Lagrangian cobordisms. We will then survey some known results, derived through Heegaard Floer Homology and contact surgery, that may provide a pathway to proving the existence of nondecomposable (nonribbon) Lagrangian cobordisms.
We generalize Ngs two-variable algebraic/combinatorial $0$-th framed knot contact homology for framed oriented knots in $S^3$ to knots in $S^1 times S^2$, and prove that the resulting knot invariant is the same as the framed cord algebra of knots. Actually, our cord algebra has an extra variable, which potentially corresponds to the third variable in Ngs three-variable knot contact homology. Our main tool is Lins generalization of the Markov theorem for braids in $S^3$ to braids in $S^1 times S^2$. We conjecture that our framed cord algebras are always finitely generated for non-local knots.
We establish an existence $h$-principle for symplectic cobordisms of dimension $2n>4$ with concave overtwisted contact boundary.
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