Computing invariants of knotted manifolds
Killeen, Kelvin
Killeen, Kelvin
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Publication Date
2022-09-13
Type
Thesis
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Abstract
In this thesis, we develop algorithms in computational topology for working with regular CW-complexes and calculating associated homological invariants. We adapt the classical notion of broken surface diagrams for describing embeddings of surfaces in 4-dimensional space, and describe algorithms for constructing regular CW-complex structures on the complements of these embedded surfaces. In particular, we develop an algorithm for inputting a virtual knot/link diagram and returning a regular CW-structure on the complement in the 4-sphere of Satoh's Tube map of the virtual knot/link. We also develop algorithms that input classical knot/link diagrams and return regular CW-structures on the complements of the knot/link in the 3-sphere as well as algorithms that perform Dehn surgery on these complements to produce any closed, orientable 3-manifold. We use homological algebra to design isotopy invariants of codimension two embeddings and describe algorithms for computing these invariants. Key illustrations of the algorithms include: - A new method for distinguishing between the granny knot and the reef knot; these knot complements have isomorphic fundamental groups and homology groups. - Recovering a calculation of Faria Martins and Kauffman that distinguishes between the spun Hopf link and the Tube of the welded Hopf link. All algorithms in this thesis are implemented in the GAP computer algebra system and are publicly available as part of the GAP package HAP.
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NUI Galway