相关论文: Foam evaluation and Kronheimer--Mrowka theories
We introduce equivariant $\mathfrak{gl}_N$ homology for links in the thickened annulus via foam evaluation.
We develop a package of monopole Floer-theoretic invariants for webs and foams using real Seiberg-Witten theory with orbifold singularities, based on Kronheimer and Mrowka's framework of monopole Floer homology.
These notes are based on the three lectures that one of the authors gave at Tsinghua University in the summer of 2023 as part of the workshop on Geometric Representation Theory and Applications. They contain an introduction to the…
We provide a finite dimensional categorification of the symmetric evaluation of $\mathfrak{sl}_N$-webs using foam technology. As an output we obtain a symmetric link homology theory categorifying the link invariant associated to symmetric…
Inspired by Kronheimer and Mrowka's approach to monopole Floer homology, we develop a model for $\mathbb{Z}/2$-equivariant symplectic Floer theory using equivariant almost complex structures, which admits a localization map to a twisted…
We introduce two invariants called sl(3) Khovanov module and pointed sl(3) Khovanov homology for spatial webs (bipartite trivalent graphs). Those invariants are related to Kronheimer-Mrowka's instanton invariants $J^\sharp$ and $I^\sharp$…
This is a survey article about knot Floer homology. We present three constructions of this invariant: the original one using holomorphic disks, a combinatorial description using grid diagrams, and a combinatorial description in terms of the…
We study equivariant coarse homology theories through an axiomatic framework. To this end we introduce the category of equivariant bornological coarse spaces and construct the universal equivariant coarse homology theory with values in the…
We consider a deformation of the Robert-Wagner foam evaluation formula, with an eye toward a relation to formal groups. Integrality of the deformed evaluation is established, giving rise to state spaces for planar GL(N) MOY graphs…
We give a purely combinatorial construction of colored $\mathfrak{sl}_n$ link homology. The invariant takes values in a 2-category where 2-morphisms are given by foams, singular cobordisms between $\mathfrak{sl}_n$ webs; applying a…
We define a "real" version of Kronheimer-Mrowka's monopole Floer homology for a 3-manifold equipped with an involution. As a special case, we obtain invariants for links via their double branched covers. The new input is the notion of a…
This article is an exposition of a body of existing results, together with an announcement of recent results. We discuss a theory of polytopes associated to bipartite graphs and trinities, developed by K\'alm\'an, Postnikov and others. This…
We construct equivariant and Bott-type Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. We present several versions of the equivariant theory:…
We proved that the dimension of the F-vector space J#(G) for a plane trivalent graph G, defined by Kronheimer and Mrowka using their SO(3) instanton Floer homology, is equal to the number of Tait colorings of G.
We survey the different versions of Floer homology that can be associated to three-manifolds. We also discuss their applications, particularly to questions about surgery, homology cobordism, and four-manifolds with boundary. We then…
The SO(3) Kauffman polynomial and the chromatic polynomial of planar graphs are categorified by a unique extension of the Khovanov homology framework. Many structural observations and computations of homologies of knots and spin networks…
We study the instanton Floer homology for links in $\mathbb{RP}^3$ and compute the second page of Kronheimer--Mrowka's spectral sequence. As an application, we show that Khovanov homology detects the unknot and the projective unknot in…
Kronheimer and Mrowka used gauge theory to define a functor $J^\sharp$ from a category of webs in $\mathbb{R}^3$ to the category of finite-dimensional vector spaces over the field of two elements. They also suggested a possible…
This is an expository paper discussing some parallels between the Khovanov and knot Floer homologies. We describe the formal similarities between the theories and give some examples which illustrate a somewhat mysterious correspondence…
We use 4-valent planar graphs and singular cobordisms (called foams) to construct an integral doubly-graded cohomology for tangles, and in particular for links, whose graded Euler characteristic yields the sl(n) link polynomial (for n > 3).