A tile model of circuit topology for self-entangled biopolymers

Abstract Building on the theory of circuit topology for intra-chain contacts in entangled proteins, we introduce tiles as a way to rigorously model local entanglements which are held in place by molecular forces. We develop operations that combine tiles so that entangled chains can be represented by algebraic expressions. Then we use our model to show that the only knot types that such entangled chains can have are $$3_1$$ 3 1 , $$4_1$$ 4 1 , $$5_1$$ 5 1 , $$5_2$$ 5 2 , $$6_1$$ 6 1 , $$6_2$$ 6 2 , $$6_3$$ 6 3 , $$7_7$$ 7 7 , $$8_{12}$$ 8 12 and connected sums of these knots. This includes all proteins knots that have thus far been identified.