Department/School

Mathematics

Date

1-18-2021

Document Type

Dataset

Keywords

Knot Theory, Petaluma, Petal Knot, Ubercrossing

Abstract

A petal projection of an (oriented) knot K consists of n (oriented) horizontal line segments (strands), each occurring at a distinct height. Looking from above at a configuration, a petal projection resembles a flower: n petals passing through one central point, where all strands cross. We label these heights 1 to n, opposite of what their z-components suggest. As we traverse the petal projection starting with the top strand (height 1), the sequence given by these heights determines a permutation (1, h_2, h_3, . . .,h_n). Thus, there is a one-to-one correspondence between n-petal projections and the (n-1)! permutations on the n-1 elements {2, 3, . . . n}.

The files in this data set provide the mapping from the above-mentioned permutations to their corresponding knot types, when the knot type has crossing number not exceeding 16. In the cases where the HOMFLYPT polynomial can distinguish between the two chiralities of a chiral knot type, we also include the chirality of the knot type. For knot types with crossing number exceeding 16, we provide an estimate of the crossing number along with an encoded version of the HOMFLYPT polynomial. Note that if one disagrees with the way we set up the heights, one can simply mirror all of the knot types.

This zipped file contains multiple plain .txt files. See the README.txt for a description of the files, the data format of the files, and how the data was computed. Specifically, the data set contains the knot types of Petaluma knots for 5, 7, 9, 11, and 13 petals.

This material is based upon work supported by the National Science Foundation under Grant No. 1720342 to Eric J. Rawdon. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Citation/Other Information

Recommended Citation: Addie McCurdy, Jason Parsley, Brandon Tran, Elizabeth Whalen, and Eric Rawdon; Knot types of Petaluma knots for 5, 7, 9, 11, and 13 petals (dataset), January 18, 2021.

Creative Commons License

Creative Commons Attribution-Noncommercial 4.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License.

Included in

Mathematics Commons

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