arXiv subjects

Samantha Allen

Publications and source records attributed to Samantha Allen.

11 recordsLinked to original sources

arXiv2026

Kleshchev multipartitions, affine Mirkovi\'c-Vilonen polytopes, and representations of KLR algebras in type ${\tt A}^{(1)}_1$

We construct explicit isomorphisms between three models for the $B(\infty)$ crystal in type ${\tt A}_1^{(1)}$: affine Mirkovi\'c--Vilonen polytopes, Kleshchev multipartitions, and a new model we call upper ledge diagrams. We also present some clarifying results on these crystals, giving a direct method for completing an affine MV polytope from the data of one of its boundary root partitions, and a non-iterative recognition theorem which characterizes Kleshchev multipartitions in type ${\tt A}_1^{(1)}$. We apply these results to the representation theory of KLR algebras, where they yield a combinatorial dictionary between cuspidal- and cellular-theoretic frameworks, along with some augmented branching rules for real root functors of induction and restriction.

math.RT↗

arXiv2026

The Host Galaxies of Active Galactic Nuclei with Direct Black Hole Mass Measurements

Reverberation mapping (RM) determines the mass of black holes (BH) in active galactic nuclei (AGNs) by resolving the BH gravitational sphere of influence in the time domain. Recent RM campaigns yielded direct BH masses through dynamical modeling for a sample of 32 objects, spanning a wide range of AGN luminosities and BH masses. In addition, accurate BH masses have been determined by spatially resolving the broad-line region with GRAVITY for a handful of AGNs. Here, we present a detailed analysis of Hubble Space Telescope images using surface-brightness profile fitting with state-of-the-art programs. We derive AGN luminosity and host-galaxy properties, such as radii and luminosities for spheroid, disk, and bar (if present). The spheroid effective radii were used to measure stellar velocity dispersion from integral-field spectroscopy. Since the BH masses of our sample do not depend on any assumption of the virial factor needed in single-epoch spectroscopic mass estimates, we can show that the resulting scaling relations between the mass of the supermassive BHs and their host galaxies match those of quiescent galaxies, naturally extending to lower masses in these (predominantly) spiral galaxies. We find that the inner AGN orientation, as traced by the broad-line region inclination angle, is uncorrelated with the host-galaxy disk. Our sample has the most direct and accurate MBH measurements of any AGN sample and provides a fundamental local benchmark for studies of the evolution of massive black holes and their host galaxies across cosmic time.

astro-ph.GA↗

arXiv2025

Knot primality: knot Floer homology, metacyclic representations and twisted homology

We develop purely algebraic methods for proving that a knot is prime. Our approach uses the Heegaard Floer polynomial in conjunction with classical knot-theoretic methods: cyclic, dihedral, and metacyclic covering spaces. The theory of twisted homology allows us to view these approaches from a unified perspective. Collectively, the primality tests developed here have proved primality for over 99.67% of knots in a large family of prime knots that includes all prime knots with 15 or fewer crossings. There are additional ways in which our approach highlights the power of Heegaard Floer methods. For one, a single computation can prove the primality of an infinite family of knots. We also illustrate the application of our approach to the setting of general three-manifolds by proving the primality of a knot in a nontrivial homology sphere.

math.GT↗

arXiv2023

Using knot Floer invariants to detect prime knots

We present knot primality tests that are built from knot Floer homology. The most basic of these is a simply stated and elementary consequence of Heegaard Floer theory: if the two-variable knot Floer polynomial of a knot K is irreducible, then K is prime. Improvements in this test yield a primality condition that has been over 90 percent effective in identifying prime knots of up to 30 crossings. As another illustration of the strength of these tools, there are 1,315 non-hyperbolic prime knots with crossing number 20 or less; the tests we develop prove the primality of over 96 percent of them. The filtered chain homotopy class of the knot Floer hat complex of a knot K has a unique minimal-dimension representative that is the direct sum of a one-dimensional complex and two-dimensional complexes, each of which can be assigned a parity. Let delta(K), b_e(K), and b_o(K) denote the dimension of this minimal representative and the number of even and odd two-dimensional summands, respectively. For a composite knot K, we observe that there is a non-trivial factorization delta(K) = mn satisfying (m-1)(n-1) \le 4 min(b_e(K), b_o(K)). This yields another knot primality test. One corollary is a simple proof of Krcatovich's result that L-space knots are prime.

math.GT↗

arXiv2022

Unknotting via null-homologous twists and multi-twists

The untwisting number of a knot K is the minimum number of null-homologous twists required to convert K to the unknot. Such a twist can be viewed as a generalization of a crossing change, since a classical crossing change can be effected by a null-homologous twist on 2 strands. While the unknotting number gives an upper bound on the smooth 4-genus, the untwisting number gives an upper bound on the topological 4-genus. The surgery description number, which allows multiple null-homologous twists in a single twisting region to count as one operation, lies between the topological 4-genus and the untwisting number. We show that the untwisting and surgery description numbers are different for infinitely many knots, though we also find that the untwisting number is at most twice the surgery description number plus 1.

math.GT↗

arXiv2022

An Upsilon torsion function for knot Floer homology

Heegaard Floer theory produces chain complexes associated to knots. Viewed as modules over polynomial rings, such complexes yield torsion invariants that offer constraints on cobordisms between knots. For instance, Juhasz, Miller and Zemke used torsion invariants to bound the number of local maxima and minima in cobordisms between pairs of knots. Gong and Marengon defined a related torsion invariant and used it to study nonorientable knot cobordisms. In this paper we define a one parameter family of Heegaard Floer torsion invariants that yields a piecewise linear function defined on the interval [0,2]. We call this the Upsilon torsion function; it is closely related to the Heegaard Floer Upsilon function defined by Ozsvath, Stipsicz and Szabo. In a natural way, this Upsilon torsion function interpolates between the Juhasz-Miller-Zemke invariant and the Gong-Marengon invariant. In addition to bounding the number of local maxima and minima in knot cobordisms, the Upsilon torsion function provides new obstructions related to the Gordian distance between knots.

math.GT↗

arXiv2020

Do Link Polynomials Detect Causality In Globally Hyperbolic Spacetimes?

Let $X$ be a $(2+1)$-dimensional globally hyperbolic spacetime with a Cauchy surface $Σ$ whose universal cover is homeomorphic to $\mathbb{R}^2$. We provide empirical evidence suggesting that the Jones polynomial detects causality in $X$. We introduce a new invariant of certain tangles related to the Conway polynomial, and prove that the Conway polynomial does not detect the connected sum of two Hopf links among relevant 3-component links, which suggests that the Conway polynomial does not detect causality in the scenario described.

math.GT↗

arXiv2020

Nonorientable surfaces bounded by knots: a geography problem

The nonorientable 4-genus is an invariant of knots which has been studied by many authors, including Gilmer and Livingston, Batson, and Ozsváth, Stipsicz, and Szabó. Given a nonorientable surface $F \subset B^4$ with $\partial F = K\subset S^3$ a knot, an analysis of the existing methods for bounding and computing the nonorientable 4-genus reveals relationships between the first Betti number $β_1$ of $F$ and the normal Euler class $e$ of $F$. This relationship yields a geography problem: given a knot $K$, what is the set of realizable pairs $(e(F), β_1(F))$ where $F\subset B^4$ is a nonorientable surface bounded by $K$? We explore this problem for families of torus knots. In addition, we use the Ozsváth-Szabó $d$-invariant of two-fold branched covers to give finer information on the geography problem. We present an infinite family of knots where this information provides an improvement upon the bound given by Ozsváth, Stipsicz, and Szabó using the Upsilon invariant.

math.GT↗

arXiv2020

Unknotting with a single twist

Given a knot in the three-sphere, is it possible to unknot it by performing a single twist, and if so, what are the possible linking numbers of such a twist? We develop obstructions to unknotting using a twist of a specified linking number. The obstructions we describe are built using classical knot invariants, Casson-Gordon invariants, and Heegaard Floer theory.

math.GT↗

arXiv2019

Concordances from differences of torus knots to $L$-space knots

It is known that connected sums of positive torus knots are not concordant to $L$-space knots. Here we consider differences of torus knots. The main result states that the subgroup of the concordance group generated by two positive torus knots contains no nontrivial $L$-space knots other than the torus knots themselves. Generalizations to subgroups generated by more than two torus knots are also considered.

math.GT↗

arXiv2017

Using secondary Upsilon invariants to rule out stable equivalence of knot complexes

Two Heegaard Floer knot complexes are called stably equivalent if an acyclic complex can be added to each complex to make them filtered chain homotopy equivalent. Hom showed that if two knots are concordant, then their knot complexes are stably equivalent. Invariants of stable equivalence include the concordance invariants $τ$, $\varepsilon$, and $Υ$. Feller and Krcatovich gave a relationship between the Upsilon invariants of torus knots. We use secondary Upsilon invariants defined by Kim and Livingston to show that these relations do not extend to stable equivalence.

math.GT↗