Coding Knots by T-Graphs

Abstract

In this paper, knots are considered as smooth embeddings of a circle into $\mathrm{ℝ}$³ defined by their flat diagrams. We propose a new method of coding knots by T-graphs describing the torsion structure on a flat diagram. For this method of coding, we introduce conceptions of a cycle and a block and describe transformations of T-graphs under the first and the third Reidemeister moves applied to the flat diagram of a knot.