zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.
57M25Knots and links in the 3-sphere
82B99Equilibrium statistical mechanics
60B99Probability theory on general structures
[1]Y. Diao, Minimal knotted polygons on the cubic lattice,J. Knot Theory Ramification 2:413–425 (1993). · Zbl 0797.57004 · doi:10.1142/S0218216593000234
[2]J. M. Hammersley, The number of polygons on a lattice,Proc. Camb. Philos. Soc. 57:516–523 (1961). · doi:10.1017/S030500410003557X
[3]H. Kesten, On the number of self-avoiding walks,J. Math. Phys. 4(7):960–969 (1963). · Zbl 0122.36502 · doi:10.1063/1.1704022
[4]N. Madras and G. Slade,The Self-Avoiding Walk (Birkhäuser, Boston, 1993).
[5]N. Pippenger, Knots in random walks,Discrete Appl. Math. 25:273–278 (1989). · Zbl 0681.57001 · doi:10.1016/0166-218X(89)90005-X
[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). · doi:10.1088/0305-4470/5/5/007
[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 · doi:10.1088/0305-4470/21/7/030