A Guide to Conway Notation

March 29, 2023

I first encountered J. Conway’s notation for knots the way I imagine most people do, by reading his original article “An enumeration of knots and links, and some of their algebraic properties.” While this was sufficient for a basic understanding, I found that the notation had evolved somewhat since 1970. I could not find a single convenient source covering common conventions, or the extensions discussed by A. Caudron in his paper “Classification des noeuds et des enlacements” (written completely in French). Finally I felt I had a great understanding of how Conway’s notation is being used, so to help others on their journey to understand the notation (and to have a convenient reference for myself) I wrote this guide.

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