Invariants of graphs in three-space

Kauffman, Louis H.

doi:10.1090/S0002-9947-1989-0946218-0

Invariants of graphs in three-space
HTML articles powered by AMS MathViewer

by Louis H. Kauffman

Trans. Amer. Math. Soc. 311 (1989), 697-710

DOI: https://doi.org/10.1090/S0002-9947-1989-0946218-0

PDF | Request permission

Abstract:

By associating a collection of knots and links to a graph in three-dimensional space, we obtain computable invariants of the embedding type of the graph. Two types of isotopy are considered: topological and rigid-vertex isotopy. Rigid-vertex 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

Embedings of

dimensional cell complexes in

determined by their

skeletons

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/S0273-0979-1985-15361-3
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/S0273-0979-1985-15304-2
Louis H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), no. 3, 395–407. MR 899057, DOI 10.1016/0040-9383(87)90009-7
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

Louis H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), no. 2, 417–471. MR 958895, DOI 10.1090/S0002-9947-1990-0958895-7
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 10.1016/0040-9383(87)90058-9

Knotentheorie

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 10.1016/0040-9383(86)90041-8
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 10.1016/0040-9383(87)90003-6

The knotting of theta curves and other graphs in

Shuji Yamada, An invariant of spatial graphs, J. Graph Theory 13 (1989), no. 5, 537–551. MR 1016274, DOI 10.1002/jgt.3190130503