Abstract by Andrea Barton
Defining invariants from petal diagrams of knots
Petal diagrams give us a convenient way to identify knots with full-length cycles in the symmetric group. In particular, a petal permutation is related to a knot projection with a single multi-crossing by describing the order in which the strands cross through the multi-crossing point. It has been shown that any two cycles which represent the same knot type can be related by a sequence of Reidemeister-type moves, which we refer to as trivial petal insertions and crossing exchanges. With these moves, we can attempt to define invariants from petal diagrams by showing that the given quantity is invariant under trivial petal additions and crossing exchanges.