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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: [email protected], [email protected]
ZACHARY ROMRELL
Affiliation:
Department of Mathematics, 18 Hoxsey St., Williams College, Williamstown, MA 01267, U.S.A. e-mails: [email protected], [email protected]
ALEXANDRA BONAT
Affiliation:
Department of Mathematics, 480 Lincoln Drive, University of Wisconsin, Madison, WI 537067, U.S.A. e-mail: [email protected]
MAYA CHANDE
Affiliation:
Department of Mathematics, Fine Hall, Washington Rd., Princeton University, Princeton, NJ 08544, U.S.A. e-mail: [email protected]
JOYE CHEN
Affiliation:
Department of Mathematics, 77 Massachusetts Ave., M.I.T., Cambridge, MA 021397, U.S.A. e-mails: [email protected], [email protected], [email protected]
MAXWELL JIANG
Affiliation:
Department of Mathematics, 77 Massachusetts Ave., M.I.T., Cambridge, MA 021397, U.S.A. e-mails: [email protected], [email protected], [email protected]
DANIEL SANTIAGO
Affiliation:
Department of Mathematics, 77 Massachusetts Ave., M.I.T., Cambridge, MA 021397, U.S.A. e-mails: [email protected], [email protected], [email protected]
BENJAMIN SHAPIRO
Affiliation:
Department of Mathematics, 27 N. Main St., Dartmouth College, Hanover, NH 03755, U.S.A. e-mail: [email protected]
DORA WOODRUFF
Affiliation:
Department of Mathematics, 1 Oxford St., Harvard University, Cambridge, MA 021387, U.S.A. e-mail: [email protected]
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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.

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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