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Generalised knotoids

Published online by Cambridge University Press:  19 September 2024

COLIN ADAMS ,
ZACHARY ROMRELL ,
ALEXANDRA BONAT ,
MAYA CHANDE ,
JOYE CHEN ,
MAXWELL JIANG ,
DANIEL SANTIAGO ,
BENJAMIN SHAPIRO  and
DORA WOODRUFF
COLIN ADAMS
Affiliation:
Department of Mathematics, 18 Hoxsey St., Williams College, Williamstown, MA 01267, U.S.A. e-mails: cadams@williams.edu, zr3@williams.edu
ZACHARY ROMRELL
Affiliation:
Department of Mathematics, 18 Hoxsey St., Williams College, Williamstown, MA 01267, U.S.A. e-mails: cadams@williams.edu, zr3@williams.edu
ALEXANDRA BONAT
Affiliation:
Department of Mathematics, 480 Lincoln Drive, University of Wisconsin, Madison, WI 537067, U.S.A. e-mail: bonat@math.wisc.edu
MAYA CHANDE
Affiliation:
Department of Mathematics, Fine Hall, Washington Rd., Princeton University, Princeton, NJ 08544, U.S.A. e-mail: mchande@princeton.edu
JOYE CHEN
Affiliation:
Department of Mathematics, 77 Massachusetts Ave., M.I.T., Cambridge, MA 021397, U.S.A. e-mails: joyec@mit.edu, mdjiang@mit.edu, dsantiag@mit.edu
MAXWELL JIANG
Affiliation:
Department of Mathematics, 77 Massachusetts Ave., M.I.T., Cambridge, MA 021397, U.S.A. e-mails: joyec@mit.edu, mdjiang@mit.edu, dsantiag@mit.edu
DANIEL SANTIAGO
Affiliation:
Department of Mathematics, 77 Massachusetts Ave., M.I.T., Cambridge, MA 021397, U.S.A. e-mails: joyec@mit.edu, mdjiang@mit.edu, dsantiag@mit.edu
BENJAMIN SHAPIRO
Affiliation:
Department of Mathematics, 27 N. Main St., Dartmouth College, Hanover, NH 03755, U.S.A. e-mail: benjamin.i.shapiro.gr@dartmouth.edu
DORA WOODRUFF
Affiliation:
Department of Mathematics, 1 Oxford St., Harvard University, Cambridge, MA 021387, U.S.A. e-mail: dorawoodruff@college.harvard.edu
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Abstract

In 2010, Turaev introduced knotoids as a variation on knots that replaces the embedding of a circle with the embedding of a closed interval with two endpoints which here we call poles. We define generalised knotoids to allow arbitrarily many poles, intervals and circles, each pole corresponding to any number of interval endpoints, including zero. This theory subsumes a variety of other related topological objects and introduces some particularly interesting new cases. We explore various analogs of knotoid invariants, including height, index polynomials, bracket polynomials and hyperbolicity. We further generalise to knotoidal graphs, which are a natural extension of spatial graphs that allow both poles and vertices.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 177 , Issue 1 , July 2024 , pp. 67 - 102
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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References

Adams, C.. Generalised augmented alternating links and hyperbolic volumes. Algebr. Geom. Topol. 17(6) (2017), 3375–3397.CrossRefGoogle Scholar
Adams, C., Bonat, A., Chande, M., Chen, J., Jiang, M., Romrell, Z., Santiago, D., Shapiro, B. and Woodruff, D.. Hyperbolic knotoids. ArXiv: 2209.04556 (2022).Google Scholar
Adams, C., Brock, J., Bugbee, J., Comar, T., Faigin, K., Huston, Amy, Joseph, A. and Pesikoff, D.. Almost alternating links. Topology Appl. 46 (1992), 151–165.CrossRefGoogle Scholar
Adams, C. and Chen, J.. Hyperbolicity of alternating links in thickened surfaces with boundary. ArXiv: 2309.04999 (2023).Google Scholar
Adams, C., Eisenberg, O., Kapoor, K., Greenberg, J., Liang, Z., O’Connor, K., Pacheco-Tallaj, N. and Wang, Y.. Turaev hyperbolicity of classical and virtual knots. Algebr. Geom. Topol. 21 (2021), 3459–3482.CrossRefGoogle Scholar
Agol, I.. The minimal volume orientable hyperbolic 2-cusped 3-manifolds. Proc. Amer. Math. Soc. 138(10) (2010), 3723–3732.CrossRefGoogle Scholar
Bhataineh, K.. An invariant of planar knotoids and finite-type invariants (2020). Preprint available at https://www.researchgate.net/publication/344446254_AN_INVARIANT_OF_PLANAR_KNOTOIDS_AND_FINITE-TYPE_INVARIANTS.Google Scholar
Bhataineh, K.. New polynomial invariants of knotoids and the theory of polar knots. Mediterranean Journal of Mathematics 19(40) (2022), 33 pages.CrossRefGoogle Scholar
Boden, H. U. and Rushworth, W.. Minimal crossing number implies minimal supporting genus. Bull. London Math. Soc. 53 (2021), 1174–1184.CrossRefGoogle Scholar
Dorier, J., Goundaroulis, D., Benedetti, F. and Stasiak, A.. Knotoid: a tool to study the entanglement of open protein chains using the concept of knotoids. Bioinformatics 34(19) (2018), 3402–3404.CrossRefGoogle Scholar
Folwaczny, L. and Kauffman, L.. A linking number definition of the affine index polynomial and applications. J. Knot Theory Ramifications 22 (1341004) (2013), 30 pages.Google Scholar
Gabrovšek, B. and Gügümcü, N.. Invariants of multi-linkoids. Mediterranean Journal of Mathematics 20 (165) (2023), 22 pages.CrossRefGoogle Scholar
Goldman, J. and Kauffman, L.. Knots, tangles and electrical networks. Adv. Math. 14 (1993), 267–306.CrossRefGoogle Scholar
Goundaroulis, D., Dorier, J., and Stasiak, A.. Knotoids and protein structure. Topol. Geom. Biopolym. 74, 2020.CrossRefGoogle Scholar
Goundaroulis, D., Gügümcü, N., Lambropoulou, S., Dorier, J., Stasiak, A. and Kauffman, L.. Topological models for open-knotted protein chains using the concepts of knotoids and bonded knotoids. Polymers 9 (2017), 444–460.CrossRefGoogle ScholarPubMed
Gügümcü, N., Gabrovsek, B. and Kauffman, L.H.. Invariants of bonded knotoids and applications to protein folding. Symmetry 14(8) (2022), 1724–1737.CrossRefGoogle Scholar
Gügümcü, N. and Kauffman, L.. New invariants of knotoids. European J. Combin. 65 (2017), 186–229.CrossRefGoogle Scholar
Gügümcü, N. and Kauffman, L.. Parity, virtual closure and minimality of knotoids. J. Knot Theory Ramifications 30 (2150076) (2022), 28 pages.Google Scholar
Gügümcü, N., Kauffman, L. and Pongtanapaisan, P.. Graphoids. ArXiv: 2209.09086 (2022).Google Scholar
Habegger, N. and Lin, X.-S.. The classification of links up to link-homotopy. J. Amer. Math. Soc. 3 (1990), 389–419.CrossRefGoogle Scholar
Henrich, A.. A sequence of degree-one vassiliev invariants for virtual knots. J. Knot Theory Ramifications 19 (2010), 461–487.CrossRefGoogle Scholar
Kashaev, R.. Invariants of long knots. In Representation Theory, Mathematical Physics and Integrable Systems, (Birkhäuser, 2019), pp. 431–451.Google Scholar
Kauffman, L.. An affine index polynomial invariant of virtual knots. Journal of Knot Theory and its Ramifications, 22 (1340007) (2013), 30 pages.Google Scholar
Kim, S., Im, Y. H. and Lee, S.. A family of polynomial invariants for knotoids. J. Knot Theory Ramifications 27 (1843001) (2018), 15 pages.Google Scholar
Kodokostas, D. and Lambropoulou, S.. Rail knotoids. J. Knot Theory Ramifications 28 (1940019) (2019), 19 pages.Google Scholar
Kutluay, D.. Winding homology of knotoids. PhD. thesis. ArXiv: 2002.07871 (2020).Google Scholar
Linov, L.. Signed heights of knotoids. J. Knot Theory Ramifications 31 (2250037) (2022), 25 pages.Google Scholar
Manturov, V. O.. Parity and projection from virtual knots to classical knots. J. Knot Theory Ramifications 22 (1350044) (2013), 20 pages.Google Scholar
Miyamoto, Y.. Volumes of hyperbolic manifolds with totally geodesic boundary. Topology 33(4) (1994), 613–629.CrossRefGoogle Scholar
Turaev, V.. Cobordism of knots on surfaces. Journal of Topology 1 (2008), 285–305.CrossRefGoogle Scholar
Turaev, V.. Knotoids. Osaka J. Math. 49(02) (2010), 195–223.Google Scholar