on whether a Conway notation describes a unique knot, so it'd be two knots which are fundamentally topologically different
2026-03-26 21:35:02.1774560902
Is there an example of two topologically different knots with the exact same Conway notation?
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Let's specify that two (unoriented) knots $K$ and $K'$ in $S^3$ are topologically "the same" iff there is a homeomorphism between $(S^3,K)$ and $(S^3,K')$ that preserves the orientation of $S^3$.
Conway notation is designed as a way to describe an (unoriented) prime knot. Prime knots generally have very many Conway notations that describe them. A particular Conway notation describes just a single prime knot. If two prime knots each can be described by the same Conway notation, then they are topologically the same.
The reason for this is that Conway notation gives a textual description of how to construct a knot diagram.
One problem with the notation is that it requires having on hand a catalog of all the basic polyhedra, with their names, vertex orders, and vertex orientations. This catalog is somewhat arbitrarily constructed, and it depends on everyone agreeing on a convention for Conway notation to be decodable. You could say that you're trading needing to have a full knot table for needing to have a table of only the alternating knots with no twists -- there are many many fewer of these.
(What fails for Conway notation for non-prime knots is that if a diagram is obviously not prime, then the basic polyhedron you get after reducing all the algebraic tangles is not in the standard list -- you're meant to describe each connect summand separately, I suppose. For the connect sum to be well-defined, you need to also somehow record orientations of each knot, and I'm not sure if Conway notation has any convention for this.)