Andrew Rupinski asked a MathOverflow question about a knot invariant which he discovered, which distinguishes the left-hand trefoil from the right-hand trefoil. Nobody’s managed to identify this invariant yet, so maybe it really is new. The construction is by colouring vertices in a stick presentation of a knot by roots of unity, which is not something I’ve seen before, and is quite simple (you can read on MO exactly what it is- there are also pictures).

Finding a new knot invariant isn’t so exciting in as of itself, but a new perspective is a valuable thing to have, and, if nothing else, Rupinski’s question does seem to present a new perspective, at least to me. MathOverflow is becoming an ever more useful resource.

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I came across a similar representation of knots as a graduate student, with vertices lying on the twisted cubic. One nice thing about this presentation is that knots are represented as permutations of {1,…,n} with a single cycle (more generally, links are represented by permutations). There is a simple “delta” move which generates link equivalence of this presentation, where you replace a stick with two sticks, if they bound a triangle which is disjoint from the other sticks (one may encode this combinatorially in terms of permutations). One can transform this representation to grid diagrams and vice versa to prove this. A grid representation of a link may be encoded as a pair of permutations, and one transforms this to a cycle representation by concatenating the permutations and changing labels. One may reverse this by applying n delta moves, so that the cycle alternates between {1,…,n} and {n+1,…,2n} (alternatively, you can think of “straightening” the twisted cubic to a line, so that each edge gets bent to an arc in a page of an open book about the line). So the cycle stick number is bounded by 2 x grid number, and vice versa.

Comment by Ian Agol — May 18, 2012 @ 12:26 pm |