Topological aspects of theta-curves in cubic lattice* *The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government Ministry of Science and ICT (NRF 2020R1G1A1A01101724, 40%). The second and third authors were supported by Institute for Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.

Abstract

Knots and embedded graphs are useful models for simulating polymer chains. In particular, a theta curve motif is present in a circular protein with internal bridges. A theta-curve is a graph embedded in three-dimensional space which consists of three edges with shared endpoints at two vertices. If we cannot continuously transform a theta-curve into a plane without intersecting its strand during the deformation, then it is said to be nontrivial. A Brunnian theta-curve is a nontrivial theta-curve that becomes a trivial knot if any one edge is removed. In this paper we obtain qualitative results of these theta-curves, using the lattice stick number which is the minimal number of sticks glued end-to-end that are necessary to construct the theta-curve type in the cubic lattice. We present lower bounds of the lattice stick number for nontrivial theta-curves by 14, and Brunnian theta-curves by 15.