## APPLIED KNOT THEORY WORKSHOP 2020

## October 09, 10am-1pm EST

## 9am-12pm CST

## 2pm-5pm GMT/UTC

#### Chris Soteros

Characterizing the entanglements in lattice models of ring polymers.

Motivated in part by recent experimental and molecular dynamics studies of the entanglement characteristics of DNA in nanonchannels, we have been studying the statistics of knotting and linking for equilibrium lattice models of polymers confined to lattice tubes. In this talk I will review our theorems and transfer-matrix-based numerical results for the knot statistics and knot localization of self-avoiding polygon models in small tubes. These results have recently been extended by Jeremy Eng to slightly larger tube sizes than previously reported (namely 2x2, 4x1, 5x1 and 3x2). The trends previously observed for smaller tube sizes continue to hold for these tube sizes. In particular we observe two modes of knotting (2-filament and 1-filament) in all tube sizes and our numerical evidence indicates that the 2-filament mode is more probable. These same two modes of knotting have been observed by others both in DNA experiments and in molecular dynamics simulations. Finally, I will present recent results and open questions about link statistics for pairs of polygons which span a lattice tube.

#### Kumar Rajeev

Topological effects in polymers.

In this talk, I will present our on-going work related to understanding topological effects in melts of rings and trefoil knotted polymers. The talk will include synthesis and modeling work in understanding topological effects in polymer melts. For the modeling work, issues of Gauge invariance in the field theory of polymers will be discussed. Also, coarse-grained molecular dynamics simulation results will be presented showing the effects of polymer chain topology in affecting disorder-order transition in diblock copolymers.

#### Dawn Ray

The number of oriented rational links with a given deficiency number.

Let Un be the set of un-oriented and rational links with crossing number n, a precise formula for |Un| was obtained by Ernst and Sumners in 1987. In this paper, we study the enumeration problem of oriented rational links. Let Mn be the set of oriented rational links with crossing number n and let Mn(d) be the set of oriented rational links with crossing number n (n ≥ 2) and deficiency d. In this paper, we derive precise formulas for $|\lambda_n|$ and $|\lambda_n(d)|$ for any given $n$ and $d$ and show that

$$

|\lambda_n(d)|=F_{n-d-1}^{(d)}+\frac{1+(-1)^{nd}}{2}F^{(\lfloor \frac{d}{2}\rfloor)}_{\lfloor \frac{n}{2}\rfloor -\lfloor \frac{d+1}{2}\rfloor},

$$

where $F_n^{(d)}$ is the convolved Fibonacci sequence.

#### Sofia Lambropoulou

Finite type invariants of knotoids.

In this talk extend the theory of finite type invariants for knots to knotoids. For spherical knotoids we show that there are non-trivial type 1 invariants, in contrast with classical knot theory where type 1 invariants vanish. We give a complete theory of type 1 invariants for spherical knotoids, by classifying linear chord diagrams of order one, and we present examples arising from the affine index polynomial and the extended bracket polynomial.

#### Eleni Panagiotou

Knot Polynomials of open and closed curves.

In this talk we introduce a method to measure entanglement of curves in 3-space that extends the notion of knot and link polynomials to open curves. We define the bracket polynomial of curves in 3-space and show that it has real coefficients and is a continuous function of the curve coordinates. This is used to define the Jones polynomial in a way that it is applicable to both open and closed curves in 3-space. For open curves, the Jones polynomial has real coefficients and it is a continuous function of the curve coordinates and as the endpoints of the curve tend to coincide, the Jones polynomial of the open curve tends to that of the resulting knot. For closed curves, it is a topological invariant, as the classical Jones polynomial. We show how these measures attain a simpler expression for polygonal curves and provide a finite form for their computation in the case of polygonal curves of 3 and 4 edges.

#### Quensisha Baldwin

The topological free energy of viral glycoproteins.

Many viruses infect cells by using a mechanism that involves binding of a viral protein to the host cell. During this process, the three-dimensional conformation of the viral binding protein changes significantly. Disruption of this process could be achieved by targeting key locations in the viral protein that are essential in this rearrangement. In this manuscript we propose to use the local geometry/topology of the crystal structure of the viral protein backbone alone to identify these essential locations. Our results show that the local Writhe, local Average Crossing Number and the local Torsion alone can identify “exotic” residues that may be essential in the viral protein infection mechanism. We apply this method to the SARS-Cov-2 Spike protein to propose target residues for drug discovery.