Invariants of graphs in threespace
HTML articles powered by AMS MathViewer
 by Louis H. Kauffman PDF
 Trans. Amer. Math. Soc. 311 (1989), 697710 Request permission
Abstract:
By associating a collection of knots and links to a graph in threedimensional space, we obtain computable invariants of the embedding type of the graph. Two types of isotopy are considered: topological and rigidvertex isotopy. Rigidvertex graphs are a category mixing topological flexibility with mechanical rigidity. Both categories constitute steps toward models for chemical and biological networks. We discuss chirality in both rigid and topological contexts.References

J. Boyle, Embedings of $2$dimensional cell complexes in ${S^3}$ determined by their $1$skeletons, (preprint).
 J. H. Conway and C. McA. Gordon, Knots and links in spatial graphs, J. Graph Theory 7 (1983), no. 4, 445–453. MR 722061, DOI 10.1002/jgt.3190070410
 P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett, and A. Ocneanu, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 2, 239–246. MR 776477, DOI 10.1090/S027309791985153613
 W. Graeub, Die semilinearen Abbildungen, S.B. Heidelberger Akad. Wiss. Math.Nat. Kl. 1950 (1950), 205–272 (German). MR 0042709
 Vaughan F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 103–111. MR 766964, DOI 10.1090/S027309791985153042
 Louis H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), no. 3, 395–407. MR 899057, DOI 10.1016/00409383(87)900097
 Louis H. Kauffman, New invariants in the theory of knots, Amer. Math. Monthly 95 (1988), no. 3, 195–242. MR 935433, DOI 10.2307/2323625 —, On knots, Ann. of Math. Studies, no. 115. Princeton Univ. Press, 1987.
 Louis H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), no. 2, 417–471. MR 958895, DOI 10.1090/S00029947199009588957
 Louis H. Kauffman and Pierre Vogel, Link polynomials and a graphical calculus, J. Knot Theory Ramifications 1 (1992), no. 1, 59–104. MR 1155094, DOI 10.1142/S0218216592000069
 Kenneth C. Millett, Stereotopological indices for a family of chemical graphs, J. Comput. Chem. 8 (1987), no. 4, 536–548. MR 892422, DOI 10.1002/jcc.540080434
 Robert D. Brandt, W. B. R. Lickorish, and Kenneth C. Millett, A polynomial invariant for unoriented knots and links, Invent. Math. 84 (1986), no. 3, 563–573. MR 837528, DOI 10.1007/BF01388747
 Kunio Murasugi, Jones polynomials and classical conjectures in knot theory, Topology 26 (1987), no. 2, 187–194. MR 895570, DOI 10.1016/00409383(87)900589 K. Reidemeister, Knotentheorie, Ergebnisse der Matematik und ihrer Grenzgebiete, (Alte Folge), Band 1, Heft 1, Springer 1932; reprint, SpringerVerlag, 1974.
 K. Reidemeister, Knot theory, BCS Associates, Moscow, Idaho, 1983. Translated from the German by Leo F. Boron, Charles O. Christenson and Bryan A. Smith. MR 717222
 Jonathan Simon, Topological chirality of certain molecules, Topology 25 (1986), no. 2, 229–235. MR 837623, DOI 10.1016/00409383(86)900418
 Jonathan Simon, Molecular graphs as topological objects in space, J. Comput. Chem. 8 (1987), no. 5, 718–726. MR 891813, DOI 10.1002/jcc.540080516
 Morwen B. Thistlethwaite, A spanning tree expansion of the Jones polynomial, Topology 26 (1987), no. 3, 297–309. MR 899051, DOI 10.1016/00409383(87)900036 K. Wolcott, The knotting of theta curves and other graphs in ${S^3}$, Thesis, Univ. of Iowa, 1985.
 Shuji Yamada, An invariant of spatial graphs, J. Graph Theory 13 (1989), no. 5, 537–551. MR 1016274, DOI 10.1002/jgt.3190130503
Additional Information
 © Copyright 1989 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 311 (1989), 697710
 MSC: Primary 57M25; Secondary 05C10
 DOI: https://doi.org/10.1090/S00029947198909462180
 MathSciNet review: 946218