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The number of smallest knots on the cubic lattice. (English) Zbl 0834.57004
Polygons on the cubic lattice are piecewise linear simple closed curves such that each linear piece is of unit length and its vertices are on the cubic lattice. Such polygons are used to simulate the behavior of (thick) random closed curves found in models from chemistry and physics. Knotting of such polygons is an interesting problem. It has been shown that a polygon on the cubic lattice needs at least 24 edges to form a knot. It is shown that the only knots one can obtain with 24 edges are the trefoils. Furthermore, by classifying the projections of the polygons on a plane, we are able to enumerate all possible knotted polygons on the cubic lattice with 24 edges. The number of such unrooted knots on the cubic lattice is 3496.
Reviewer: Y.Diao (Marietta)

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
82B99 Equilibrium statistical mechanics
60B99 Probability theory on algebraic and topological structures
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[1] Y. Diao, Minimal knotted polygons on the cubic lattice,J. Knot Theory Ramification 2:413–425 (1993). · Zbl 0797.57004
[2] J. M. Hammersley, The number of polygons on a lattice,Proc. Camb. Philos. Soc. 57:516–523 (1961). · Zbl 0122.36501
[3] H. Kesten, On the number of self-avoiding walks,J. Math. Phys. 4(7):960–969 (1963). · Zbl 0122.36502
[4] N. Madras and G. Slade,The Self-Avoiding Walk (Birkhäuser, Boston, 1993). · Zbl 0780.60103
[5] N. Pippenger, Knots in random walks,Discrete Appl. Math. 25:273–278 (1989). · Zbl 0681.57001
[6] M. F. Sykes, D. S. McKenzie, M. G. Watts, and J. L. Martin, The number of self-avoiding rings on a lattice,J. Phys. A: Gen. Phys. 5:661–666 (1972).
[7] D. W. Sumners and S. G. Whittington, Knots in self-avoiding walks,J. Phys. A: Math. Gen. 21:1689–1694 (1988). · Zbl 0659.57003
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