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Zbl 0797.57004
Diao, Yuanan
Minimal knotted polygons on the cubic lattice. (English)
[J] J. Knot Theory Ramifications 2, No.4, 413-425 (1993). ISSN 0218-2165

Summary: The polygons on the cubic lattice have played an important role in simulating various circular molecules, especially the ones with relatively big volumes. There have been a lot of theoretical studies and computer simulations devoted to this subject. The questions are mostly around the knottedness of such a polygon, such as what kind of knots can appear in a polygon of given length, how often it can occur, etc. A very often asked and long standing question is about the minimal length of a knotted polygon. It is well-known that there are knotted polygons on the lattice with 24 steps yet it is unproved up to this date that 24 is the minimal number of steps needed. In this paper, we prove that in order to obtain a knotted polygon on the cubic lattice, at least 24 steps are needed and we can only have trefoils with 24 steps.

MSC 2000:: *57M25 Knots and links in the 3-sphere

Keywords: selfavoiding walks; polygons on the cubic lattice; circular molecules; knots; minimal length of a knotted polygon; trefoils

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