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Colorado State University

1. Eddy, Thomas D. Improved stick number upper bounds.

Degree: MS(M.S.), Mathematics, 2019, Colorado State University

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 R3. 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 935, 939, 943, 945, and 948. In addition, the best-known stick number upper bounds were improved for 60 other knots with crossing number ten and below. Advisors/Committee Members: Shonkwiler, Clayton (advisor), Adams, Henry (committee member), Chitsaz, Hamid (committee member).

Subjects/Keywords: knot theory; stick number; toric symplectic manifold; polygon index; edge number; symplectic geometry

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APA (6th Edition):

Eddy, T. D. (2019). Improved stick number upper bounds. (Masters Thesis). Colorado State University. Retrieved from

Chicago Manual of Style (16th Edition):

Eddy, Thomas D. “Improved stick number upper bounds.” 2019. Masters Thesis, Colorado State University. Accessed July 20, 2019.

MLA Handbook (7th Edition):

Eddy, Thomas D. “Improved stick number upper bounds.” 2019. Web. 20 Jul 2019.


Eddy TD. Improved stick number upper bounds. [Internet] [Masters thesis]. Colorado State University; 2019. [cited 2019 Jul 20]. Available from:

Council of Science Editors:

Eddy TD. Improved stick number upper bounds. [Masters Thesis]. Colorado State University; 2019. Available from: