Petal Projections, Knot Colorings and Determinants

An \"{u}bercrossing diagram is a knot diagram with only one crossing that may involve more than two strands of the knot. Such a diagram without any nested loops is called a petal projection. Every knot has a petal projection from which the knot can be recovered using a permutation that represents strand heights. Using this permutation, we give an algorithm that determines the \(p\)-colorability and the determinants of knots from their petal projections. In particular, we compute the determinants of all prime knots with crossing number less than \(10\) from their petal permutations.