Related papers: The cube and the Burnside category
Suppose given a Frobenius category E, i.e. an exact category with a big enough subcategory B of bijectives. Let_E_ := E/B denote its classical stable category. For example, we may take E to be the category of complexes C(A) with entries in…
We create a framework for odd Khovanov homology in the spirit of Bar-Natan's construction for the ordinary Khovanov homology. Namely, we express the cube of resolutions of a link diagram as a diagram in a certain 2-category of chronological…
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) representations. Projective functors…
We consider a pivotal monoidal functor whose domain is a modular tensor category (MTC). We show that the trace of such a functor naturally extends to a representation of the corresponding tube category. As irreducible representations of the…
We show that a variation of Milnor's $\bar\mu$-invariants, the so-called Campbell-Hausdorff invariants introduced recently by Stefan Papadima, are of finite type with respect to {\it marked singular links}. These link invariants are…
We study configuration space integral formulas for Milnor's homotopy link invariants, showing that they are in correspondence with certain linear combinations of trivalent trees. Our proof is essentially a combinatorial analysis of a…
We define an annular concordance invariant and study its properties. When specialized to braids, this invariant gives bounds on band rank. We introduce a modified chain complex to reformulate the invariant. Then, by focusing on a special…
Link Floer homology is an invariant for links which has recently been described entirely in a combinatorial way. Originally constructed with mod 2 coefficients, it was generalized to integer coefficients thanks to a sign refinement. In this…
The aim of this paper is to define two link invariants satisfying cubic skein relations. In the hierarchy of polynomial invariants determined by explicit skein relations they are the next level of complexity after Jones, HOMFLY, Kauffman…
We derive an inductive, combinatorial definition of a polynomial-valued regular isotopy invariant of links and tangled graphs. We show that the invariant equals the Reshetikhin-Turaev invariant corresponding to the exceptional simple Lie…
We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl(2) and sl(3) and by…
We introduce a multivariable Casson-Lin type invariant for links in $S^3$. This invariant is defined as a signed count of irreducible $\operatorname{SU}(2)$ representations of the link group with fixed meridional traces. For 2-component…
This paper is devoted to the study of algebraic structures leading to link homology theories. The originally used structures of Frobenius algebra and/or TQFT are modified in two directions. First, we refine 2-dimensional cobordisms by…
Link homotopy has been an active area of research for knot theorists since its introduction by Milnor in the 1950s. We introduce a new equivalence relation on spatial graphs called component homotopy, which reduces to link homotopy in the…
The homology cobordism group of homology cylinders is a generalization of the mapping class group and the string link concordance group. We study this group and its filtrations by subgroups by developing new homomorphisms. First, we define…
We develop a homotopy theory for additive categories endowed with endofunctors, analogous to the concept of a model structure. We use it to construct the homotopy theory of a Hovey triple (which consists of two compatible complete cotorsion…
This paper examines and strengthens the Cuntz-Thomsen picture of equivariant Kasparov theory for arbitrary second-countable locally compact groups, in which elements are given by certain pairs of cocycle representations between C*-dynamical…
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 the category of cobordisms of oriented…
We define and study the properties of observables associated to any link in $\Sigma\times {\bf R}$ (where $\Sigma$ is a compact surface) using the combinatorial quantization of hamiltonian Chern-Simons theory. These observables are traces…
We define an invariant of based transverse links, as a well-defined element inside the equivariant Heegaard Floer cohomology of its branched double cover, defined by Lipschitz, Hendricks, and Sarkar. We prove the naturality and…