arXiv subjects

Melody Molander

Publications and source records attributed to Melody Molander.

6 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↗

arXiv2026

NIM-representations of Tambara-Yamagami generalizations

We compute and classify the irreducible non-negative integer matrix (NIM-)representations of two proposed generalizations of the Tambara-Yamagami fusion ring, as studied by Jordan-Larson and Galindo-Lentner-M\"oller, respectively. We also detect the algebra objects associated to these NIM-representations.

math.QA↗

arXiv2026

A mirror deformation of Markov numbers

We introduce a deformed squared Markov equation given by $X^2 + Y^2 + Z^2 + (q+q^{-1})(XY+YZ+XZ) = 3(1 + q + q^{-1})XYZ$. Symmetric solutions of this new equation present a remarkable factorization property which allows us to talk about their square roots. These square roots give a natural $q$-deformation of the Markov numbers that has not previously occurred in the literature. We call them mirror Markov numbers. We prove a characterization of mirror Markov numbers and discover a mutation rule, mirror mutation, to generate them all. We also prove a geometric realization of the corresponding mirror mutation on a once-punctured sphere with three orbifold points. Our mirror deformation leads to deformations of Fibonacci and Pell branches for which we give precise formulas. Furthermore, the deformed squared Markov equation specializes to many other very well known generalized Markov equations. We also obtain the super Markov numbers from a specialization of the deformed squared Markov numbers, which we use to prove a conjecture of Musiker.

math.CO↗

arXiv2026

A Well-Defined Jellyfish Algorithm for the Affine $E_7$ Subfactor Planar Algebra

In this paper, we contribute to the Kuperberg program by giving a diagrammatic presentation of generators and relations for the affine $E_7$ unshaded subfactor planar algebra. Using this presentation, we prove that its jellyfish algorithm is a well-defined surjection onto $\mathbb{C}$. In particular, this shows that the jellyfish algorithm is an invariant on closed diagrams for this planar algebra.

math.QA↗

arXiv2024

Skein Theory for Affine A Subfactor Planar Algebras

The Kuperberg Program asks to find presentations of planar algebras and use these presentations to prove results about their corresponding categories purely diagrammatically. This program has been completed for index less than 4 and is ongoing research for index greater than 4. We give generators-and-relations presentations for the affine A subfactor planar algebras of index 4. Exclusively using the planar algebra language, we give new proofs to how many such planar algebras exist. Categories corresponding to these planar algebras are monoidally equivalent to cyclic pointed fusion categories. We give a proof of this by defining a functor yielding a monoidal equivalence between the two categories. The categories are also monoidally equivalent to a representation category of a cyclic subgroup of SU(2). We give a new proof of this fact, explicitly using the diagrammatic presentations found. This gives novel diagrammatics for these representation categories.

math.QA↗

arXiv2024

A determinant formula of the Jones polynomial for a family of braids

In 2012, Cohen, Dasbach, and Russell presented an algorithm to construct a weighted adjacency matrix for a given knot diagram. In the case of pretzel knots, it is shown that after evaluation, the determinant of the matrix recovers the Jones polynomial. Although the Jones polynomial is known to be #P-hard by Jaeger, Vertigan, and Welsh, this presents a class of knots for which the Jones polynomial can be computed in polynomial time by using the determinant. In this paper, we extend these results by recovering the Jones polynomial as the determinant of a weighted adjacency matrix for certain subfamilies of the braid group. Lastly, we compute the Kauffman polynomial of (2,q) torus knots in polynomial time using the balanced overlaid Tait graphs. This is the first known example of generalizing the methodology of Cohen to a class of quantum invariants which cannot be derived from the HOMFLYPT polynomial.

math.GT↗