arXiv subjects

Louisa Liles

Publications and source records attributed to Louisa Liles.

5 recordsLinked to original sources

arXiv2026

Thompson's Group $V$ and Virtual Link Theory

Thompson's groups $F \subset T \subset V$ were introduced in 1965 and have since found widespread application in fields as diverse as logic, group theory, homotopy theory, and lattice gauge theory. In 2014, V. F. R. Jones constructed unitary representations of $F$, factoring through a surjection from $F$ to isotopy classes of links in $S^3$. The second author extended Jones' surjection to $T$, thereby constructing all isotopy classes of checkerboard colorable (CC) links in the thickened annulus. We complete this program for $V$, defining a surjection $\mathcal{L}_{V}$ from $V$ to virtual equivalence classes of CC links in thickened compact oriented surfaces. This yields a new oriented subgroup $\vec{V} \subset V$ containing Jones' oriented subgroups $\vec{F} \subset F$ and $\vec{T}\subset T$. We prove $\vec{V}$ realizes all oriented almost classical virtual links. We then construct unitary representations of $V$ and $\vec{V}$ from kei and operator quandle coloring invariants, respectively.

math.GT↗

arXiv2024

$(t,q)$-Series Invariants of Seifert Manifolds

Gukov, Pei, Putrov, and Vafa developed a $q$-series invariant of negative definite plumbed $3$-manifolds with spin$^{c}$ structures, building on earlier work of Lawrence and Zagier. This was recently generalized to an an infinite family of two-variable $(t,q)$-series invariants by Akhmechet, Johnson, and Krushkal (AJK). We calculate one such series for all Seifert manifolds with $b_{1}=0.$ These results extend a previous theorem of Liles and McSpirit to any number of exceptional fibers and the Reduction Theorem of Gukov, Svoboda, and Katzarkov to the two-variable case. As a consequence, a previous result of Liles and McSpirit on modularity properties and radial limits is enhanced to a larger family of manifolds. We also calculate the infinite collection of $(t,q)$-series invariants for three infinite families of manifolds, finding mixed modularity properties for one such family.

math.GT↗

arXiv2024

Annular Links from Thompson's Group $T$

In 2014 Jones showed how to associate links in the $3$-sphere to elements of Thompson's group $F$. We provide an analogue of this program for annular links and Thompson's group $T$. The main result is that any edge-signed graph embedded in the annulus is the Tait graph of an annular link built from an element of $T$. In analogy to the work of Aiello and Conti, we also show that the coefficients of certain unitary representations of $T$ recover the Jones polynomial of annular links.

math.GT↗

arXiv2024

Thompson's group $F$, tangles, and link homology

We extend a construction of Jones to associate $(n, n)$-tangles with elements of Thompson's group $F$ and prove that it is asymptotically faithful as $n \to\infty$. Using this construction we show that the oriented Thompson group $\vec F$ admits a lax group action on a category of Khovanov's chain complexes.

math.GT↗

arXiv2023

Infinite Families of Quantum Modular 3-Manifold Invariants

One of the first key examples of a quantum modular form, which unifies the Witten-Reshetikhin-Turaev (WRT) invariants of the Poincaré homology sphere, appears in work of Lawrence and Zagier. We show that the series they construct is one instance in an infinite family of quantum modular invariants of negative definite plumbed 3-manifolds whose radial limits toward roots of unity may be thought of as a deformation of the WRT invariants. We use a recently developed theory of Akhmechet, Johnson, and Krushkal (AJK) which extends lattice cohomology and BPS $q$-series of 3-manifolds. As part of this work, we provide the first calculation of the AJK series for an infinite family of $3$-manifolds. Additionally, we introduce a separate but related infinite family of invariants which also exhibit quantum modularity properties.

math.GT↗