TL;DR
This paper reviews the classification and tools related to closed 3-braids, highlighting their significance as a manageable yet rich class of knots and links for testing conjectures and understanding braid index 3.
Contribution
It compiles and reviews tools specific to knots and links of braid index 3, providing a comprehensive resource for calculations and understanding exceptional cases.
Findings
Complete classification of closed 3-braids
Identification of exceptional cases in braid index 3
Open questions in the study of 3-braids
Abstract
Knots and links which are closed 3-braids are a very special class. Like 2-bridge knots and links, they are simple enough to admit a complete classification. At the same time they are rich enough to serve as a source of examples on which, with luck, a researcher may be able to test various conjectures. The goal of this review article is to gather together, in one place, some of the tools that are special to knots and links of braid index 3, in a form that could be useful for those who have a need to calculate, and need to know precisely all the exceptional cases. We also use it as an opportunity to review what is known and suggest some open questions.
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