Graduation Year


Document Type

Campus Only Senior Thesis

Degree Name

Bachelor of Science



Reader 1

Francis Edward Su

Reader 2

Dagan Karp

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2021 Max K Chao-Haft


This thesis explores possible sink-source-cycle theorems for discrete vector fields on 3-dimensional polytopal complexes. We begin with an intuitive introduction to the Poincaré-Hopf index theorem and a detailed review of L. Glass’s graph-theoretic analogue of the Poincaré-Hopf index. This transitions to a discussion of the “filled board theorem” – a sink-source-cycle theorem in two dimensions – first discovered by R. Alvarado, M. Averett, B. Gaines, C. Jackson, M.L. Karker, M.A. Marciniak, F. Su, and S. Walker and later reproved, via a discrete index theorem, by S. Ammons. We attempt to extend this result to three dimensions and analyze the obstructions we encounter.

This thesis is restricted to the Claremont Colleges current faculty, students, and staff.