Computing Invariants of Knotted Graphs Given by Sequences of Points in 3-Dimensional Space
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We design a fast algorithm for computing the fundamental group of the complement to any knotted polygonal graph in 3-space. A polygonal graph consists of straight segments and is given by sequences of vertices along edge-paths. This polygonal model is motivated by protein backbones described in the Protein Data Bank by 3D positions of atoms. Our KGG algorithm simplifies a knotted graph and computes a short presentation of the Knotted Graph Group containing powerful invariants for classifying graphs up to isotopy. We use only a reduced plane diagram without building a large complex representing the complement of a graph in 3-space.
KeywordsFundamental Group Protein Backbone Geometric Graph Isotopy Class Cubical Complex
This work was done during the EPSRC-funded secondment at Microsoft Research Cambridge, UK. We thank all reviewers for valuable comments and helpful suggestions.
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