Steenrod Squares on Khovanov Homology and an Implementation
2022-11
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Steenrod Squares on Khovanov Homology and an Implementation
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2022-11
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Link invariants allow us to distinguish links. Often, the more complicated or involved a knot invariant is to calculate the more links it can successfully help us distinguish. In this paper, we start with the Jones polynomial, which is a relatively straightforward invariant to compute. We build upon to that to define Khovanov homology, a more involved but also more informative invariant. From there, we describe Moran’s construction of the Steenrod squares sq^i on Khovanov homology. These operations allow us to further distinguish knots with identical Khovanov homology groups. We present a few by-hand calculations of these operations, which quickly become unwieldy.Then, we describe our software implementation of Steenrod squares on Khovanov homology in Sagemath. This forms a package called KhovanovSteenrodSquare and is available at https://github.com/samilstein/KhovanovSteenrodSquare. As long as a link does not contain a component that is the unknot or the Hopf link, this code can compute the Steenrod square on any of the Khovanov homology groups for that link. We used this code to compute sq^1, sq^2, sq^3, and sq^4 on the disjoint union of one, two, three, and four trefoils respectively. We also engaged in a systematic search for examples of nontrivial sq^4 for knots with up to 15 crossings. This implementation allows us to capture the usefulness of this invariant while decreasing the effort required to compute it.
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University of Minnesota Ph.D. dissertation. November 2022. Major: Mathematics. Advisor: Tyler Lawson. 1 computer file (PDF); ix, 120 pages.
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Milstein, Sarah. (2022). Steenrod Squares on Khovanov Homology and an Implementation. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/252325.
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