Generalizations of Knotoids and Spatial Graphs

Colin Adams; Alexandra Bonat; Maya Chande; Joye Chen; Maxwell Jiang; Zachary Romrell; Daniel Santiago; Benjamin Shapiro; Dora Woodruff

arXiv · 2209.01922

Generalizations of Knotoids and Spatial Graphs

Colin Adams·Alexandra Bonat·Maya Chande·Joye Chen·Maxwell Jiang·Zachary Romrell·Daniel Santiago·Benjamin Shapiro·Dora Woodruff

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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 generalized 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 generalize to knotoidal graphs, which are a natural extension of spatial graphs that allow both poles and vertices.

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BibTeX RIS

Colin Adams, Alexandra Bonat, Maya Chande, Joye Chen, Maxwell Jiang, Zachary Romrell, Daniel Santiago, Benjamin Shapiro, Dora Woodruff. 2022-09-05. Generalizations of Knotoids and Spatial Graphs. https://arxiv.org/abs/2209.01922

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