A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Simplified Khovanov polynomials for bipartite links.
Investigates polynomial time algorithms for computing Khovanov homology of braids.
Method for computing Khovanov homology of tangles.
Researchers compute Khovanov polynomials for satellite knots.
New insights into Khovanov polynomials using tangle calculus.
Explains Khovanov homology and its applications.
Khovanov homology is a categorification of the Jones polynomial, so it may be seen as a kind of quantum invariant of knots and links. Although polynomial quantum invariants are deeply involved with Vassiliev (aka. finite type) invariants, the relation remains unclear in case of Khovanov homology. Aiming at it, in this …
Paper defines new versions of Jones polynomial and Khovanov homology.
The working mathematician fears complicated words but loves pictures and diagrams. We thus give a no-fancy-anything picture rich glimpse into Khovanov's novel construction of `the categorification of the Jones polynomial'. For the same low cost we also provide some computations, including one that shows that Khovanov's…
The Jones polynomial is a famous link invariant that can be defined diagrammatically via a skein relation. Khovanov homology is a richer link invariant that categorifies the Jones polynomial. Using spectral sequences, we obtain a skein-type relation satisfied by the Khovanov homology. Thanks to this relation, we are ab…
Motivated by Khovanov homology and relations between the Jones polynomial and graph polynomials, we construct a homology theory for embedded graphs from which the chromatic polynomial can be recovered as the Euler characteristic. For plane graphs, we show that our chromatic homology can be recovered from the Khovanov h…
Proof of isotopy invariance in Khovanov link homology.
Khovanov homology is a recently introduced invariant of oriented links in . It categorifies the Jones polynomial in the sense that the (graded) Euler characteristic of the Khovanov homology is a version of the Jones polynomial for links. In this paper we study torsion of the Khovanov homology. Based on ou…
Khovanov homology offers a nontrivial generalization of Jones polynomial of links in R^3 (and of Kauffman bracket skein module of some 3-manifolds). In this chapter (Chapter X) we define Khovanov homology of links in R^3 and generalize the construction into links in an I-bundle over a surface. We use Viro's approach to…
New bounds on Khovanov homology for positive links families.
Khovanov homology extended to 3-manifolds, linking tangles.
Proves existence of colored Khovanov bicomplex linking Jones polynomial.
Two categorifications are given for the arrow polynomial, an extension of the Kauffman bracket polynomial for virtual knots. The arrow polynomial extends the bracket polynomial to infinitely many variables, each variable corresponding to an integer {\it arrow number} calculated from each loop in an oriented state summa…
New spin on Khovanov-Rozansky homology categorifies spin link polynomial.
The Jones polynomial can be expressed in terms of spanning trees of the graph obtained by checkerboard coloring a knot diagram. We show there exists a complex generated by these spanning trees whose homology is the reduced Khovanov homology. The spanning trees provide a filtration on the reduced Khovanov complex and a …
In this paper we give a quantum statistical interpretation for the bracket polynomial state sum <K> and for the Jones polynomial. We use this quantum mechanical interpretation to give a new quantum algorithm for computing the Jones polynomial. This algorithm is useful for its conceptual simplicity, and it applies to al…
This paper defines girth for knots and links, linking it to Khovanov homology.
We explain how to compute the Jones polynomial of a link from one of its grid diagrams and we observe a connection between Bigelow's homological definition of the Jones polynomial and Kauffman's definition of the Jones polynomial. Consequently, we prove that the Maslov grading on the Seidel-Smith symplectic link invari…
In the first few homological gradings, there is an isomorphism between the Khovanov homology of a link and the categorification of the chromatic polynomial of a graph related to the link. In this article, we show that the categorification of the chromatic polynomial only contains torsion of order two, and hence Khovano…
The Jones polynomial and Khovanov homology of a classical link are invariants that depend upon an initial choice of orientation for the link. In this paper, we give a Khovanov homology theory for unoriented virtual links. The graded Euler characteristic of this homology is proportional to a similarly-defined unoriented…
Study eight categorifications of colored Jones polynomial, verifying physics conjectures.
This paper explains how to compute Khovanov homology of torus links using the Kauffman bracket polynomial.
In this thesis we work with Khovanov homology of links and its generalizations, as well as with the homology of graphs. Khovanov homology of links consists of graded chain complexes which are link invariants, up to chain homotopy, with graded Euler characteristic equal to the Jones polynomial of the link. Hence, it can…
We describe an invariant of links in the three-sphere which is closely related to Khovanov's Jones polynomial homology. Our construction replaces the symmetric algebra appearing in Khovanov's definition with an exterior algebra. The two invariants have the same reduction modulo 2, but differ over the rationals. There i…
We prove that the Khovanov homology of the 2-cable detects the unknot. A corollary is that Khovanov's categorification of the 2-colored Jones polynomial detects the unknot.
Simplified Khovanov-Rozansky calculus for bipartite knots.
Jones-Wenzl projectors lifted to Khovanov spectra, proving knot conjectures.
The paper conjectures Khovanov homology can distinguish torus and twist knots.
In analogy with a recursive formula for the HOMFLY-PT polynomial of links given by Jaeger, we give a recursive formula for the graph polynomial introduced by Kauffman and Vogel. We show how this formula extends to the Khovanov-Rozansky graph homology.
Witt algebra acts on Khovanov-Rozansky homology of links.
Computes Khovanov homology for 2-strand braids via graph relations.
New insights into Khovanov homology complexity and topological structure.
Expands Jones polynomial for Legendrian knots with categorification.
Khovanov homology gaps in quasi-alternating links are shown to be one.
We present an easy example of mutant links with different Khovanov homology. The existence of such an example is important because it shows that Khovanov homology cannot be defined with a skein rule similar to the skein relation for the Jones polynomial.
This paper is an introduction to Khovanov homology, starting with the Kauffman bracket state summation, emphasizing the Bar-Natan Canopoloy and tangle cobordism approach. The paper discusses a simplicial approach to Khovanov homology and a quantum model for it so that the graded Euler characteristic that produces the J…
Derives Khovanov homology for 2-strand braids using combinatorial relations.
Khovanov homology detects essential surfaces in knot complements.
Khovanov homology is a bigraded Z-module that categorifies the Jones polynomial. The support of Khovanov homology lies on a finite number of slope two lines with respect to the bigrading. The Khovanov width is essentially the largest horizontal distance between two such lines. We show that it is possible to generate in…
In this paper we compute the signature for a family of knots , the weaving knots of type . By work of E.~S.~Lee the signature calculation implies a vanishing theorem for the Khovanov homology of weaving knots. Specializing to knots , we develop recursion relations that enable us to compute the Jo…
We show a spectral sequence for the rational Khovanov homology of an oriented link in terms of the rational Khovanov complexes and homologies of the link surgeries along an admissible cut. As a non trivial corollary, we give an explicit splitting formula for the Jones polynomial.
This article provides an overview of relative strengths of polynomial invariants of knots and links, such as the Alexander, Jones, Homflypt, Kaufman two-variable polynomial, and Khovanov polynomial.
Quantum algorithm approximates Khovanov homology ranks.