RNA foldings and Stuck Knots

TL;DR

This paper introduces a novel approach combining knot theory and stuck knots to better model RNA foldings, capturing both entanglement and intrachain interactions, and provides algebraic invariants for analysis.

Contribution

It develops a new algebraic framework and invariants for stuck knots, specifically applied to RNA folding diagrams, enhancing topological modeling of biomolecules.

Findings

Defined a generating set of stuck Reidemeister moves.

Introduced an algebraic structure for stuck links.

Computed the coloring invariant for RNA foldings.

Abstract

We study RNA foldings and investigate their topology using a combination of knot theory and embedded rigid vertex graphs. Knot theory has been helpful in modeling biomolecules, but classical knots place emphasis on a biomolecule's entanglement while ignoring their intrachain interactions. We remedy this by using stuck knots and links, which provide a way to emphasize both their entanglement and intrachain interactions. We first give a generating set of the oriented stuck Reidemeister moves for oriented stuck links. We then introduce an algebraic structure to axiomatize the oriented stuck Reidemeister moves. Using this algebraic structure, we define a coloring counting invariant of stuck links and provide explicit computations of the invariant. Lastly, we compute the counting invariant for arc diagrams of RNA foldings through the use of stuck link diagrams.