Matrix presentations of protein knotting. Each entry in the matrix indicates the knot type formed by one continuous subchain by its shading () or color (–). In each case, the subchain starts with the N-terminal amino acid at position *x* and ends with the C-terminal amino acid at position *y*, indicated on the horizontal and vertical axes, respectively. Equivalently, this subchain can be interpreted as a part of the diagonal, delimited by the corresponding coordinates *x* and *y*, where the entire diagonal corresponds to the entire polypeptide chain. (*A*) A case of a knotted protein. Notice that the entry in the lower left-hand corner, which corresponds to the entire protein, is shaded; this indicates that the corresponding chain or subchain is knotted. The knot core is defined as the shortest subchain that still forms a knot (see the thickened part of the protein in the sketch above). The two remaining parts of the chain form knot tails, and their length is conveniently represented along the diagonal. Subchains that do not include at least short bits of both knot tails do not form knots and therefore the matrix entries corresponding to these subchains are not shaded. (*B* and *C*) Cases of protein slipknots. Notice that in the case of slipknots the entire protein is unknotted (the element in the lower corner is white) but as one (*B*) or both termini (*C*) are trimmed to some extent, the remaining fragment forms a trefoil knot, denoted 3_{1} in and *SI Appendix*, Table S2, and the corresponding matrix entries are therefore darkened in the matrix. Schematic drawings of the polypeptide chains forming trefoil knots and slipknots illustrate which parts of the polypeptide chains constitute the knot cores (thickened), the knot and slipknot tails (solid line), and the slipknot loops (dashed line).

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