Computing Invariants of Knotted Graphs Given by Sequences of Points in 3-Dimensional Space
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.
- 2.Crowell, R., Fox, R.: Introduction to Knot Theory. Graduate Texts in Mathematics, vol. 57. Springer, New York (1963)Google Scholar
- 3.Ellis, G.: HAP — Homological Algebra Programming package for GAP. Version 1.10.13 (2013). Available for download at http://www.gap-systems.org/Packages/hap.html
- 6.Jamroz, M., Niemyska, W., Rawdon, E., Stasiak, A., Millett, K., Sulkowski, P., Sulkowska, J.: KnotProt: a database of proteins with knots and slipknots. Nucleic Acids Res. 1, 1–9 (2014)Google Scholar
- 12.Kurlin, V.: A linear time algorithm for visualizing knotted structures in 3 pages. In: Proceedings of IVAPP: Information Visualization Theory and Applications, Berlin, pp. 5–16 (2015)Google Scholar
- 14.Kurlin, V., Smithers, C.: A linear time algorithm for embedding arbitrary knotted graphs into a 3-page book. In: Computer Vision, Imaging and Computer Graphics Theory and Applications, pp. 99–122. Springer, Berlin (2016). http://www.springer.com/gb/book/9783319299709
- 19.Waldhausen, F.: On irreducible 3-manifolds which are sufficiently large. Ann. Math. (2) 87, 56–88 (1968)Google Scholar