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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. 

How knotted proteins fold has remained controversial since the identification of deeply knotted proteins nearly two decades ago. Both computational and experimental approaches have been used to investigate protein knot formation. Motivated by the computer simulations of Bölinger et al. [Bölinger D, et al. (2010)PLoS Comput Biol6:e1000731] for the folding of the $6_1$-knotted α-haloacid dehalogenase (DehI) protein, we introduce a topological description of knot folding that could describe pathways for the formation of all currently known protein knot types and predicts knot types that might be identified in the future. We analyze fingerprint data from crystal structures of protein knots as evidence that particular protein knots may fold according to specific pathways from our theory. Our results confirm Taylor’s twisted hairpin theory of knot folding for the $3_1$-knotted proteins and the $4_1$-knotted ketol-acid reductoisomerases and present alternative folding mechanisms for the $4_1$-knotted phytochromes and the $5_2$- and $6_1$-knotted proteins.