相关论文: Foam evaluation and Kronheimer--Mrowka theories
This survey explores knot polynomials and their categorification, culminating in the homological invariants of knots. We begin with an overview of classical knot polynomials, progressing towards the superpolynomial and its role in unifying…
Equivariant singular instanton Floer theory is a framework that associates to a knot in an integer homology 3-sphere a suite of homological invariants that are derived from circle-equivariant Morse-Floer theory of a Chern-Simons functional…
We establish a structural understanding of the involutive Heegaard Floer homology for all linear combinations of almost-rational (AR) plumbed three-manifolds. We use this to show that the Neumann-Siebenmann invariant is a homology cobordism…
Let $(F,J,\omega)$ be an almost K\"ahler manifold, $\alpha$ a $J$-holomorphic action of a compact Lie group $\hat K$ on $F$, and $K$ a closed normal subgroup of $\hat K$ which leaves $\omega$ invariant. We introduce gauge theoretical…
In this note, we describe a connection between the enumerative geometry of curves in K3 surfaces and the chiral ring of an auxiliary superconformal field theory. We consider the invariants calculated by Yau--Zaslow (capturing the Euler…
For any negative definite plumbed 3-manifold M we construct from its plumbed graph a graded Z[U]-module. This, for rational homology spheres, conjecturally equals the Heegaard-Floer homology of Ozsvath and Szabo, but it has even more…
We determine the structure of the equivariant cohomology and $K$-theory of Bott towers. By restriction, we obtain similar results for Bott-Samelson varieties. This results allow us to describe more precisely the equivariant cohomology and…
We associate a moduli problem to a colored trivalent graph; such graphs, when planar, appear in the state-sum description of the quantum sl(N) knot polynomial due to Murakami, Ohtsuki, and Yamada. We discuss how the resulting moduli space…
In this paper, we present a constructive and proof-relevant development of graph theory, including the notion of maps, their faces, and maps of graphs embedded in the sphere, in homotopy type theory. This allows us to provide an elementary…
Certain foams and relations on them are introduced to interpret functors and natural transformations in categories of representations of iterated wreath products of cyclic groups of order two. We also explain how patched surfaces with…
Gromov-Witten invariants of weighted projective planes and Euler characteristics of moduli spaces of representations of bipartite quivers are related via the tropical vertex, a group of formal automorphisms of a torus. On the Gromov-Witten…
The Pfaffian structure of the boundary monomer correlation functions in the dimer-covering planar graph models is rederived through a combinatorial / topological argument. These functions are then extended into a larger family of…
This paper explores the conjecture that the following are equivalent for rational homology 3-spheres: having left-orderable fundamental group, having non-minimal Heegaard Floer homology, and admitting a co-orientable taut foliation. In…
In this article, we introduce equivariant formal deformation theory of Lie triple systems. We introduce an equivariant deformation cohomology of Lie triple systems and using this we study the equivariant formal deformation theory of Lie…
We show that the conformal blocks constructed in the previous article by the first and the third author may be described as certain integrals in equivariant cohomology. When the bundles of conformal blocks have rank one, this construction…
We introduce and study elementary properties of graph homology of algebras. This new homology theory shares many features of cyclic and Hochschild homology. We also define a graph K-theory together with an analog of Chern character.
We discuss generalizations of Ozsvath-Szabo's spectral sequence relating Khovanov homology and Heegaard Floer homology, focusing attention on an explicit relationship between natural Z (resp., 1/2 Z) gradings appearing in the two theories.…
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 pictures a little longer than in other…
We give a combinatorial characterization of generic minimal rigidity for planar periodic frameworks. The characterization is a true analogue of the Maxwell-Laman Theorem from rigidity theory: it is stated in terms of a finite combinatorial…
We investigate the combinatorial analogues, in the context of normal surfaces, of taut and transversely measured (codimension 1) foliations of 3-manifolds. We establish that the existence of certain combinatorial structures, a priori weaker…