Abstract
A method for partitioning topologically chiral knots into mutually heterochiral classes has been developed, based on the principle that for such knots there exist no diagrams whose vertex-bicolored graphs are composed of equivalent black and white subgraphs. The method, which introduces the concept ofwrithe profiles, is successfully applied to alternating as well as non-alternating prime and composite knots, and works in cases where the Jones and Kauffman polynomials fail to recognize the knot's chirality. It is shown that writhe profiles are sensitive indicators of diagram similarity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
W.T. Kelvin,Baltimore Lectures on Molecular Dynamics and the Wave Theory of Light (C.J. Clay, London, 1904) p. 619.
E. Ruch, Theor. Chim. Acta (Berl.) 11 (1968) 183;
E. Ruch, Acc. Chem. Res. 5 (1972) 49.
S.J. Tauber, J. Res. Nat. Bur. Stand. 67A (1963) 591.
K. Murasugi, Topology 26 (1987) 187;
[4a]M.B. Thistlethwaite, Topology 27 (1988) 311.
E. Flapan, Topological techniques to detect chirality, in:New Developments in Molecular Chirality, ed. P.G. Mezey (Kluwer Acad. Publ., Dordrecht, 1991) pp. 209–239.
V. Prelog and G. Helmchen, Angew. Chem. Int. Ed. Engl. 21 (1982) 567, and references therein.
G. Schill,Catenanes, Rotaxanes, and Knots (Academic Press, New York, 1971) p. 18.
D.M. Walba, Tetrahedron 41 (1985) 3161.
D. Rolfsen,Knots and Links (Publish or Perish, Berkeley, 1976; second printing with corrections: Publish or Perish, Houston, 1990), Appendix C: Table of knots and links, pp. 388–429.
M.B. Thistlethwaite, unpublished results.
C. Liang and Y. Jiang, J. Theor. Biol. 158 (1992) 231.
C. Liang and K. Mislow, J. Math. Chem. 15 (1994) 1.
K. Mislow and M. Raban, Stereoisomeric relationships of groups in molecules, in:Topics in Stereochemistry,Vol. 1, eds. N.L. Allinger and E.L. Eliel (Wiley, New York, 1967) pp. 1–38; K. Mislow and J. Siegel, J. Amer. Chem. Soc. 106 (1984) 3319.
O. Frank,Statistical inference in Graphs (Foa Repro, Stockholm, 1971) pp. 26–30.
T.A. Brown, J. Combin. Theory 1 (1966) 498;
F. Harary and E.M. Palmer,Graphical Enumeration (Academic Press, New York, 1973) pp. 231–233.
G. Burde and H. Zieschang,Knots (Walter de Gruyter, Berlin, 1985), fig. 14.12, p. 265.
P.G. Tait, On knots I, II, III,Scientific Papers Vol. I (Cambridge University Press, London, 1898) pp. 273–347. Plates V and VII.
R.H. Crowell and R.H. Fox,Introduction to Knot Theory (Blaisdell, New York, 1963) p. 132.
J.H. Conway, An enumeration of knots and links, and some of their algebraic properties, in:Computational Problems in Abstract Algebra, ed. J. Leech (Pergamon Press, New York, 1970) pp. 329–358; fig. 8, p. 335.
For an overview, see: K.C. Millett, Algebraic topological indices of molecular chirality, in:New Developments in Molecular Chirality, ed. P.G. Mezey (Kluwer Acad. Publ., Dordrecht, 1991) pp. 165–207.
M.B. Thistlethwaite, Knot tabulations and related topics, in:Aspects of Topology, London Math. Soc. Lecture Note Series no. 93, eds. I.M. James and E.H. Kronheimer (Cambridge University Press, Cambridge, 1985) pp. 1–76.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Liang, C., Mislow, K. A left-right classification of topologically chiral knots. J Math Chem 15, 35–62 (1994). https://doi.org/10.1007/BF01277547
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01277547