Topological aspects of theta-curves in cubic lattice*

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.

Knots and θ-Curves Identification in Polymeric Chains and Native Proteins Using Neural Networks

“…However, while a closed knot is a circle embedded in 3 , a θ-curve is a spatial graph that consists of three edges with two common vertices. 50 For this reason, we proposed changing the shape of the features to…”

Lattice conformation of theta-curves accompanied with Brunnian property

“…In the rest of this section, we naturally assume that a lattice Brunnian θ-curve Θ is irreducible and properly leveled. In the preceding paper [20] by the authors, the following result was proved. From now on, we only consider the case that Θ consists of exactly 15 sticks, and so |Θ| x = |Θ| y = |Θ| z = 5.…”

“…Recently, a general upper bound of the lattice stick number of θ-curves was found by Yoo et al in [32]. This work is a sequel to the research on finding better lower bounds of the lattice stick number for Brunnian theta-curves [20]. A Brunnian theta-curve Θ is a nontrivial θ-curve in which each pair of three knotoids making up Θ becomes the trivial knot.…”

“…In the proof they used the fact that the three-connected sum of two Brunnian θ-curves is also Brunnian [31], and two properties which are the superadditivity of τ under the three-connected sum [20] and Scharlemann and Thompson's result regarding maximal Euler characteristics of oriented links [23]. Note that Kinoshita θ-curve θ5 1 realizes τ (θ5 1 ) = 2.…”

“…Figure 2 shows the Kinoshita θ-curve which is a typical Brunnian theta-curve and its lattice presentation. In the previous paper [20], the authors showed that at least 14 and 15 lattice sticks are needed to construct nontrivial θ-curves and Brunnian theta-curves, respectively. In this paper, we improve the lower bound of the lattice stick number for Brunnian theta-curves.…”