An official website of the United States government
Abstract We describe the universal target of annular Khovanov–Rozansky link homology functors as the homotopy category of a free symmetric monoidal linear category generated by one object and one endomorphism. This categorifies the ring of symmetric functions and admits categorical analogues of plethystic transformations, which we use to characterize the annular invariants of Coxeter braids. Further, we prove the existence of symmetric group actions on the Khovanov–Rozansky invariants of cabled tangles and we introduce spectral sequences that aid in computing the homologies of generalized Hopf links. Finally, we conjecture a characterization of the horizontal traces of Rouquier complexes of Coxeter braids in other types.
more »
« less
This content will become publicly available on June 1, 2026
Roots of Alexander polynomials of random positive 3-braids
Motivated by an observation of Dehornoy, we study the roots of Alexander polynomials of knots and links that are closures of positive 3-strand braids. We give experimental data on random such braids and find that the roots exhibit marked patterns, which we refine into precise conjectures. We then prove several results along those lines, for example that generically at least 69% of the roots are on the unit circle, which appears to be sharp. We also show there is a large root-free region near the origin. We further study the equidistribution properties of such roots by introducing a Lyapunov exponent of the Burau representation of random positive braids, and a corresponding bifurcation measure. In the spirit of Deroin and Dujardin, we conjecture that the bifurcation measure gives the limiting measure for such roots, and prove this on a region with positive limiting mass. We use tools including work of Gambaudo and Ghys on the signature function of links, for which we prove a central limit theorem.
more »
« less
- Award ID(s):
- 2303572
- PAR ID:
- 10598122
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Advances in Mathematics
- Volume:
- 471
- Issue:
- C
- ISSN:
- 0001-8708
- Page Range / eLocation ID:
- 110254
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
abstract: In this manuscript we study braid varieties, a class of affine algebraic varieties associated to positive braids. Several geometric constructions are presented, including certain torus actions on braid varieties and holomorphic symplectic structures on their respective quotients. We also develop a diagrammatic calculus for correspondences between braid varieties and use these correspondences to obtain interesting decompositions of braid varieties and their quotients. It is shown that the maximal charts of these decompositions are exponential Darboux charts for the holomorphic symplectic structures, and we relate these charts to exact Lagrangian fillings of Legendrian links.more » « less
-
Many well studied knots can be realized as positive braid knots where the braid word contains a positive full twist; we say that such knots are twist positive. Some important families of knots are twist positive, including torus knots, 1-bridge braids, algebraic knots, and Lorenz knots. We prove that if a knot is twist positive, the braid index appears as the third exponent in its Alexander polynomial. We provide a few applications of this result. After observing that most known examples of L-space knots are twist positive, we prove: if K is a twist positive L-space knot, the braid index and bridge index of K agree. This allows us to provide evidence for Baker’s reinterpretation of the slice-ribbon conjecture: that every smooth concordance class contains at most one fibered, strongly quasipositive knot. In particular, we provide the first example of an infinite family of positive braid knots which are distinct in concordance, and where, as g tends to infinity, the number of hyperbolic knots of genus g gets arbitrarily large. Finally, we collect some evidence for a few new conjectures, including the following: the braid and bridge indices agree for any L-space knot.more » « less
-
null (Ed.)Abstract We investigate the topologies of random geometric complexes built over random points sampled on Riemannian manifolds in the so-called “thermodynamic” regime. We prove the existence of universal limit laws for the topologies; namely, the random normalized counting measure of connected components (counted according to homotopy type) is shown to converge in probability to a deterministic probability measure. Moreover, we show that the support of the deterministic limiting measure equals the set of all homotopy types for Euclidean connected geometric complexes of the same dimension as the manifold.more » « less
-
We investigate the uniform reshuffling model for money exchanges: two agents picked uniformly at random redistribute their dollars between them. This stochastic dynamics is of mean-field type and eventually leads to a exponential distribution of wealth. To better understand this dynamics, we investigate its limit as the number of agents goes to infinity. We prove rigorously the so-called propagation of chaos which links the stochastic dynamics to a (limiting) nonlinear partial differential equation (PDE). This deterministic description, which is well-known in the literature, has a flavor of the classical Boltzmann equation arising from statistical mechanics of dilute gases. We prove its convergence toward its exponential equilibrium distribution in the sense of relative entropy.more » « less
