A stick knot is a mathematical knot formed by a chain of straight line segments. For a knot K, define the stick number of K, denoted stick(K), to be the minimum number of straight edges necessary to form a stick knot which is equivalent to K. Stick number is a knot invariant whose precise value is unknown for the large majority of knots, although theoretical and observed bounds exist.

There is a natural correspondence between stick knots and polygons in three-dimensional Euclidean space. Previous research has attempted to improve observed stick number upper bounds by computationally generating such polygons and identifying the knots that they form. This thesis presents a new variation on this method which generates equilateral polygons in tight confinement, thereby increasing the incidence of polygons forming complex knots. Our generation strategy is to sample from the space of confined polygons by leveraging the toric symplectic structure of this space. An efficient sampling algorithm based on this structure is described.

This method was used to discover the precise stick number of knots 9_35, 9_39, 9_43, 9_45, and 9_48. In addition, the best-known stick number upper bounds were improved for 60 other knots with crossing number ten and below. 

The stick number of a knot is the minimum number of straight edges needed to construct the knot. This invariant is unknown for most knots, although various theoretical and observed bounds are known. Previous research has attempted to improve the observed upper bounds for stick number by generating random polygons and identifying the knots they form. This talk will present a new variation on this method which generates equilateral polygons in tight confinement to increase the incidence of complex knots. Even in tight confinement, the formation of complex knots is rare and thus one must choose a generation algorithm which is capable of producing large numbers of samples rapidly. This talk will describe such an algorithm, which is based on the toric symplectic structure of equilateral polygon space, and how to use it to generate and identify the knot type of billions of polygons. The talk will conclude with new stick number upper bounds for more than 20 knots, including exact stick numbers for two knots, obtained using this method.

The study of symplectic manifolds arose from the formulation of Hamiltonian mechanics. This talk will cover some of the foundational definitions and concepts from symplectic geometry. We will also work through several examples which emphasize the intuition behind these ideas. In particular, we focus on understanding and interpreting Hamiltonian actions and the resulting moment maps. 

This talk will begin with an introduction to symplectic geometry. In particular, we will define and give examples of toric symplectic manifolds and discuss some important results related to these constructions. We will then relate these findings to a particular toric symplectic manifold, the space of stick knots modulo rigid rotation. With this context in mind, the latter half of the talk will deal with a knot invariant called the stick number, which is the minimal number of straight edges which can be chained together to form a given knot. The talk will conclude with new results which improve the stick number upper bound of many knots, including the discovery of the precise stick number for two knots. 

This talk will present an introduction to the mathematical theory of knots with a focus on knot invariants. In particular, we will focus on the invariant called stick number, the minimal number of straight edges which can be chained together to form a given knot. The precise stick number is unknown for most knots although various theoretical and observed bounds exist. The talk will conclude with new results which improve the stick number upper bound of many knots, including the discovery of the precise stick number for two knots.

Slides from this talk are available here.