Sam Nelson

In this talk I will recount the history of my knot theory-based music project and show an example of my method for creating music from knot homsets.

M. Niebrzydowski and J. H. Przytycki defined a Kauffman bracket magma and constructed the invariant P of framed links in 3-space. The invariant is closely related to the Kauffman bracket […]

(Joint with Ichihara, Jong, and Saito). We show that two-bridge knots admit no purely cosmetic surgeries, ie no pair of distinct Dehn surgeries on such a knot produce 3-manifolds that […]

We talk about building knots using mosaics which were as introduced as a way of modeling quantum knots by Lomonaco and Kauffman and a newer variant, hexagonal mosaics, introduced by […]

The Jones polynomial is an invariant of knots in R^3. Following a proposal of Witten, it was extended to knots in 3-manifolds by Reshetikhin-Turaev using quantum groups. Khovanov homology is […]

Skein modules were introduced by Jozef H. Przytycki as generalisations of the Jones and HOMFLYPT polynomial link invariants in the 3-sphere to arbitrary 3-manifolds. The Kauffman bracket skein module (KBSM) […]

Convex real projective structures generalize hyperbolic structures in a rich way. We will discuss a class of manifolds introduced by Cooper Long and Tillmann, which include finite-volume cusped hyperbolic manifolds […]

The non-orientable 4-genus of a knot K is the smallest first Betti number of any non-orientable surface in the 4-ball spanning the knot. It is defined to be zero if […]

Title: Chebyshev Threadings in Skein Algebras for Punctured Surfaces Abstract: Skein algebras are algebras of links in a surface quotiented by diagram-based equivalence relations based on the Kauffman bracket. In […]

Title: Cellular resolutions of the diagonal and exceptional collections for toric Deligne-Mumford stacks Abstract: Beilinson gave a resolution of the diagonal for complex projective space which yields a strong, full exceptional […]

Title: Cellular resolutions of the diagonal and exceptional collections for toric Deligne-Mumford stacks (Continued) Abstract: Beilinson gave a resolution of the diagonal for complex projective space which yields a strong, full exceptional collection of line bundles. Bayer-Popescu-Sturmfels generalized Beilinson's result to a cellular resolution of the diagonal for what they called "unimodular" toric varieties (a more […]

Title: Quantum 4-Manifold Invariants Via Trisections Abstract: I will describe a new family of potentially non-semisimple invariants for compact a 4-manifold whose boundary is equipped with an open book. The invariant is computed using a trisection, along with some additional combing data, and a piece of algebraic data called a Hopf triple. The relationship with […]