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.
We give a geometric interpretation of Bar-Natan's universal invariant for the class of tangles in the 3-ball with four ends: we associate with such 4-ended tangles T multicurves BN(T), that is, collections of immersed curves with local systems in the 4-punctured sphere. These multicurves …
A non-separating multicurve of a surface S of genus g with m punctures is a multicurve c so that S-c is connected. For k>0 define the graph of non-separting k-multicurves to be the graph whose vertices are non-separating multicurves with k components and where two such multicurves are connected by an edge if they can b…
We study some graphs associated to a surface, called k-multicurve graphs, which interpolate between the curve complex and the pants graph. Our main result is that, under certain conditions, simplicial embeddings between multicurve graphs are induced by π1-injective embeddings of the corresponding surfaces. We also p…
We present an algorithm for calculating the geometric intersection number of two multicurves on the n-punctured disk, taking as input their Dynnikov coordinates. The algorithm has complexity O(m2n4), where m is the sum of the absolute values of the Dynnikov coordinates of the two multicurves. The main ingredien…
We propose an elementary model to price European physical delivery swaptions in multicurve setting with a simple exact closed formula. The proposed model is very parsimonious: it is a three-parameter multicurve extension of the two-parameter Hull-White (1990) model. The model allows also to obtain simple formulas for a…
Study of random multicurves and square-tiled surfaces on large genus surfaces.
problem Understanding the geometry and combinatorial properties of random multicurves and square-tiled surfaces on surfaces of large genus.
method Combination of combinatorial and geometric analysis, including large genus asymptotic analysis of moduli space volumes and intersection numbers.
result Random multicurves and square-tiled surfaces have well-approximated properties by random permutations, with specific expected values.
Given a set equipped with a transitive action of a group, we define the notion of an almost invariant coloring of the set. We consider the mapping class group orbit of a multicurve on a compact surface, and prove that in the case of genus at least two, no such almost invariant coloring exists. Conversely, in the case o…
After Fossas-Parlier, we consider two graphs G0(S) and G∞(S), constructed from multicurves on connected, orientable surfaces of infinite-type. Our first result asserts that G∞(S) has finite diameter, which extends a result of Fossas-Parlier. Next, we prove that the…
We consider the representation space of a compact surface, that is the space of morphisms from the fundamental group to SU(2) up to conjugation. We show that the trace functions associated to multicurves on the surface are linearly independent as functions on the representation space. The proof relies on the Fourier de…
Various curve complexes with vertices representing multicurves on a surface S have been defined, for example [3], [4] and [8]. The homology curve complex HC(S,α) defined in [7] is one such complex, with vertices corresponding to multicurves in a nontrivial integral homology class α. Given two multicurve…
In this paper, we determine the distribution of the length partition of a random multicurve of fixed topological type on a closed hyperbolic surface using the methods of Margulis' thesis and Mirzakhani's equidistribution theorem for horospheres. This distribution admits a polynomial density, whose coefficients can be e…
A \textit{multicurve} $\C$ on a closed orientable surface is defined to be a finite collection of disjoint non-isotopic essential simple closed curves. The Dehn twist $t_{\C}$ about $\C$ is the product of the Dehn twists about the individual curves. In this paper, we give necessary and sufficient conditions for the exi…
Let X be a closed hyperbolic surface and λ,η be weighted geodesic multicurves which are short on X. We show that the iterated grafting along λ and η is close in the Teichmueller metric to grafting along a single multicurve which can be given explicitly in terms of λ and η. Using this result, we study the h…
We partially solve the conjecture by A.Shumakovitch about torsion in the Khovanov homology of prime, non-split links in S^3. We give a size restriction on the Khovanov homology of almost alternating links. We relate the Khovanov homology of the connected sum of a link diagram and the Hopf link with the Khovanov homolog…
We introduce Khovanov homology for ribbon graphs and show that the Khovanov homology of a certain ribbon graph embedded on the Turaev surface of a link is isomorphic to the Khovanov homology of the link (after a grading shift). We also present a spanning quasi-tree model for the Khovanov homology of a ribbon graph.
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…
On a surface with a Finsler metric, we investigate the asymptotic growth of the number of closed geodesics of length less than L which minimize length among all geodesic multicurves in the same homology class. An important class of surfaces which are of interest to us are hyperbolic surfaces.