A Reidemeister type theorem for petal diagrams of knots

Abstract

We study petal diagrams of knots, which provide a method of describing knots in terms of permutations in a symmetric group S 2n+1 . We define two classes of moves on such permutations, called trivial petal additions and crossing exchanges, which do not change the isotopy class of the underlying knot. We prove that any two permutations which represent isotopic knots can be related by a sequence of these moves and their inverses.

Petal Projections, Knot Colorings and Determinants

“…Since we can begin traveling around the petal projection at any petal, recording the heights of the strands we encounter along the way, we can cyclically permute the permutation that describes the projection. For example, the trefoil in Figure 7 is defined by the permutation (1,3,5,2,4), but it could also be defined as (3,5,2,4,1) or (5,2,4,1,3). For this reason, we follow the convention in [1] to always write petal permutations starting with the 1.…”

“…This is equivalent to making the highest strand the lowest, the second highest the second lowest, and so on. Therefore, the trefoil could also be written: (5, 3, 1, 4, 2) (and then shifted to (1,4,2,5,3)). Notice that while 1 is at the start of both this representation and the original one, they are still two different petal permutations describing the same knot.…”

“…Furthermore, since we can always move the bottom strand around to the top of the pile of übercrossing strands, we can start recording the order of the strands at any height. So, the permutation (1, 6, 4, 2, 5, 7, 3) could also be written as (2, 7, 5, 3, 6, 1, 4) by adding 1 to each entry (mod 7), which can be rewritten as (1,4,2,7,5,3,6).…”

“…This alone, with the properties discussed above, shows that the trefoil is the only knot with petal number 5. Indeed, there exists a more complete classification of when two petal permutations describe the same knot, given in the recent paper A Reidemeister type theorem for petal diagrams of knots [3]. This is an exciting development, as it gives more credence to the theory that petal projections provide a new, useful language from the geometric to the purely algebraic.…”

“…Code for testing the algorithm on sets of petal projections #Here, we import the code from Listing 1 and an algorithm timing library in Python. import spp_library import timeit petal_projections_to_test = [[1,3,5,2,4],[1,3,5,2, 7,4, 6],[1,3,5, 2, 8, 4, 6, → 9, 7]] number_of_runs_each = 100 def mean(runtimes): return sum(runtimes) / len(runtimes) for petal_projection in petal_projections_to_test: #Compute the knot determinant petal_projection_determinant = spp_library.knotDeterminant(petal_projection) alg_runtimes = [] #Run the petal projection calculation number_of_runs_each times and get a runtime → list for i in range(number_of_runs_each): start = timeit.default_timer() spp_library.knotDeterminant(petal_projection) alg_runtimes.append(1000*(timeit.default_timer() -start))…”