Shortest paths of Rubik’s Snake prime knots up to 5 crossings

Abstract

A Rubik’s Snake is a toy that was invented over 40 years ago together with the more famous Rubik’s Cube. It can be twisted into many interesting shapes including knots. Four blocks can form a trivial knot. In this paper, we study how many blocks are needed to form a nontrivial knot with up to 5 crossings. The results are classified using the DT (Dowker-Thistlethwaite) code to make sure each design is indeed the knot we claimed it is. A line representation is used to clearly reveal the knot structure of the Rubik’s Snake. Exhaustive local searches are performed to verify that no local improvement is possible for the shortest paths we found.