New York Journal of Mathematics
Volume 26 (2020), 836-852


Erica Flapan, Kenji Kozai, and Ryo Nikkuni

Stick number of non-paneled knotless spatial graphs

view    print

Published: July 29, 2020.
Keywords: Non-paneled knotless spatial graphs, stick embeddings of graphs, metalloproteins, Mobius ladders, ravels, K3,3, K4, K5.
Subject: 57M15, 57K10, 05C10, 92C40, 92E10.

We show that the minimum number of sticks required to construct a non-paneled knotless embedding of K4 is 8 and of K5 is 12 or 13. We use our results about K4 to show that the probability that a random linear embedding of K3,3 in a cube is in the form of a Mobius ladder is 0.97380 ± 0.00003, and offer this as a possible explanation for why K3,3 subgraphs of metalloproteins occur primarily in this form.


The first author was supported in part by NSF Grant DMS-1607744. The third author was supported by JSPS KAKENHI Grant Number JP15K04881.

Author information

Erica Flapan:
Department of Mathematics
610 N. College Ave.
Pomona College
Claremont, CA 91711, USA


Kenji Kozai:
Department of Mathematics
Rose-Hulman Institute of Technology
5500 Wabash Ave., Terre Haute, IN 47803, USA


Ryo Nikkuni:
Department of Mathematics
Tokyo Woman's Christian University
2-6-1 Zempukuji, Suginami-ku, Tokyo 167-8585, Japan