Related papers: A new computational tool for Khovanov cobordism ma…
The checkerboard coloring of knot diagrams offers a graph-theoretical approach to address topological questions. Champanerkar and Kofman defined a complex generated by the spanning trees of a graph obtained from the checkerboard coloring…
We introduce a diagrammatic approach to Rasmussen's $s$-invariant, based on Bar-Natan's reformulation of Khovanov homology for tangles and cobordisms. This method enables a local computation of $s$ from a tangle decomposition of a knot…
A theorem of Kronheimer and Mrowka states that Khovanov homology is able to detect the unknot. That is, if a knot has the Khovanov homology of the unknot, then it is equivalent to it. Similar results hold for the trefoils and the…
We describe algorithms for computing geometric invariants for Hilbert modular surfaces, and we report on their implementation.
We give a new method for calculating the cohomology of the normal bundles over rational varieties which are smooth projections of Veronese embeddings. The method can be used also when the projections are not smooth, in this case it provides…
We describe a method for generating minimal hard prime surface-link diagrams. We extend the known examples of minimal hard prime classical unknot and unlink diagrams up to three components and generate figures of all minimal hard prime…
Khovanov homology is a recently introduced invariant of oriented links in $\mathbb{R}^3$. It categorifies the Jones polynomial in the sense that the (graded) Euler characteristic of the Khovanov homology is a version of the Jones polynomial…
We compute some cohomology spaces for the symmetric power of the tautological bundle tensor the determinant bundle on the punctual Hilbert scheme of a complex smooth projective surface.
We provide a diagrammatic computation for the bilinear form, which is defined as the pairing between the (relative) cup products with every local coefficients and every integral homology 2-class of every links in the 3-sphere. As a…
A second part of detailed elementary introduction into Khovanov homologies. This part is devoted to reduced Jones superpolynomials. The story is still about a hypercube of resolutions of a link diagram. Each resolution is a collection of…
Khovanov homology ist a new link invariant, discovered by M. Khovanov, and used by J. Rasmussen to give a combinatorial proof of the Milnor conjecture. In this thesis, we give examples of mutant links with different Khovanov homology. We…
We show that bordered Heegaard Floer homology detects incompressible surfaces and bordered-sutured Floer homology detects partly boundary parallel tangles and bridges, in natural ways. For example, there is a bimodule Lambda so that the…
We introduce Hopf algebroid covariance on Woronowicz's differential calculus. Using it, we develop quite a general framework of noncommutative complex geometry that subsumes the one in [2]. We present transverse complex and K\"ahler…
This paper begins with a survey of some applications of Khovanov homology to low-dimensional topology, with an eye toward extending these results to $\mathfrak{sl}(n)$ homologies. We extend Levine and Zemke's ribbon concordance obstruction…
The study of the intersection cohomology of moduli spaces of semistable bundles was initiated by Frances Kirwan in the 1980's. In this paper, we give a complete geometric proof of a recursive formula, which reduces the calculation of the…
We obtain new multilinear multiplier theorems for symbols of restricted smoothness which lie locally in certain Sobolev spaces. We provide applications concerning the boundedness of the commutators of Calder\'on and…
We study factorizations of HOMFLY polynomials of certain knots and oriented links. We begin with a computer analysis of knots with at most 12 crossings, finding 17 non-trivial factorizations. Next, we give an irreducibility criterion for…
While the problem of computing the genus of a knot is now fairly well understood, no algorithm is known for its four-dimensional variants, both in the smooth and in the topological locally flat category. In this article, we investigate a…
Using region crossing changes, we define a new invariant called the multi-region index of a knot. We prove that the multi-region index of a knot is bounded from above by twice the crossing number of the knot. In addition, we show that the…
We introduce a family of periods of mixed Tate motives called dissection polylogarithms, that are indexed by combinatorial objects called dissection diagrams. The motivic coproduct on the former is encoded by a combinatorial Hopf algebra…