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Constrained knots in lens spaces

Fan Ye

Algebraic & Geometric Topology 23 (2023) 1097–1166
Abstract

We study a special family of (1,1) knots called constrained knots, which includes 2–bridge knots in the 3–sphere S3 and simple knots in lens spaces. Constrained knots are parametrized by five integers and characterized by the distribution of spinc structures in the corresponding (1,1) diagrams. The knot Floer homology HFK^ of a constrained knot is thin. We obtain a complete classification of constrained knots based on the calculation of HFK^ and presentations of knot groups. We provide many examples of constrained knots constructed from surgeries on links in S3, which are related to 2–bridge knots and 1–bridge braids. We also show many examples of constrained knots whose knot complements are orientable hyperbolic 1–cusped manifolds with simple ideal triangulations.

Keywords
$(1,1)$ knots, $2$–bridge knots, Heegaard Floer homology, SnapPy manifolds
Mathematical Subject Classification
Primary: 57K10, 57K14, 57K18, 57K31, 57K32
References
Publication
Received: 8 July 2020
Revised: 1 July 2021
Accepted: 13 September 2021
Published: 6 June 2023
Authors
Fan Ye
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
Cambridge
United Kingdom

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