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ALMOST UNKNOTTED EMBEDDINGS OF GRAPHS IN Z³ AND HIGHER DIMENSIONAL ANALOGUES

N. MADRAS

Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada

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D. W. SUMNERS

Department of Mathematics, Florida State University, Tallahassee, Florida, USA

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, and

S. G. WHITTINGTON

Department of Chemistry, University of Toronto, Toronto, Ontario, Canada

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https://doi.org/10.1142/S0218216509007336Cited by:2 (Source: Crossref)

Abstract

We consider the number of almost unknotted embeddings of graphs in Z³. We show that the number of such embeddings is the same, to exponential order, as the number of unknotted embeddings. We also consider some higher dimensional analogues, ie almost unknotted embeddings of surfaces which are p-dimensional analogues of Θ-graphs in Z^p+2. We describe a lattice version of the spinning construction which establishes the embeddability of such surfaces in Z^p+2 and show that the number of embeddings is the same, to exponential order, as the number of unknotted embeddings. The proofs of our upper bounds feature a novel application of the classical Loomis–Whitney inequality.

Keywords:

AMSC: 57M25, 57Q45, 82B41, 26D15

References

Y. Diao, J. Knot Theory Ramifications 2, 413 (1992), DOI: 10.1142/S0218216593000234. Link, Web of Science, Google Scholar
Y. Diao, J. Knot Theory Ramifications 4, 189 (1995), DOI: 10.1142/S0218216595000090. Link, Web of Science, Google Scholar
Y. Diao, J. C. Nardo and Y. Sun, J. Knot Theory Ramifications 10, 597 (2001), DOI: 10.1142/S0218216501001025. Link, Web of Science, Google Scholar
Y. Diao, N. Pippenger and D. W. Sumners, J. Knot Theory Ramifications 3, 419 (1994), DOI: 10.1142/S0218216594000307. Link, Google Scholar
E. Flapan , When Topology Meets Chemistry ( Cambridge University Press , Cambridge , 2000 ) . Crossref, Google Scholar
J. M. Hammersley, Proc. Cambridge Phil. Soc. 57, 516 (1961), DOI: 10.1017/S030500410003557X. Crossref, Google Scholar
D. Jungreis, J. Knot Theory Ramifications 3, 455 (1994), DOI: 10.1142/S0218216594000332. Link, Google Scholar
A. Kawauchi, Osaka J. Math. 26, 743 (1989). Web of Science, Google Scholar
H. Kesten, J. Math. Phys. 4, 960 (1963), DOI: 10.1063/1.1704022. Crossref, Web of Science, Google Scholar
S. Kinoshita, Pacific J. Math. 42, 89 (1972). Crossref, Web of Science, Google Scholar
S. Kinoshita, Pacific J. Math. 49, 127 (1973). Crossref, Web of Science, Google Scholar
S. Kinoshita, Pacific J. Math. 57, 217 (1975). Crossref, Web of Science, Google Scholar
S. Kinoshita, Topology and Computer Science 211 (1987). Google Scholar
L. H. Loomis and H. Whitney, Bull. Amer. Math. Soc. 55, 961 (1949), DOI: 10.1090/S0002-9904-1949-09320-5. Crossref, Web of Science, Google Scholar
N. Madras and G. Slade , The Self-Avoiding Walk ( Birkhäuser , Boston , 1993 ) . Google Scholar
W. K. Mason, Trans. Amer. Math. Soc. 142, 269 (1969), DOI: 10.2307/1995358. Web of Science, Google Scholar
J. McAtee, D. S. Silver and S. G. Williams, J. Knot Theory Ramifications 10, 109 (2001), DOI: 10.1142/S0218216501000755. Link, Web of Science, Google Scholar
N. Pippenger, Discrete Appl. Math. 25, 273 (1989), DOI: 10.1016/0166-218X(89)90005-X. Crossref, Web of Science, Google Scholar
D. Rolfsen , Knots and Links ( Publish or Perish, Inc. , Wilmington , 1976 ) . Google Scholar
J. K. Simon and K. Wolcott, Topol. Appl. 37, 163 (1990), DOI: 10.1016/0166-8641(90)90061-6. Crossref, Web of Science, Google Scholar
C. E. Soteros, J. Phys. A: Math. Gen. 25, 3153 (1992), DOI: 10.1088/0305-4470/25/11/023. Crossref, Google Scholar
C. E. Soteros, D. W. Sumners and S. G. Whittington, Math. Proc. Cambridge Phil. Soc. 111, 75 (1992), DOI: 10.1017/S0305004100075174. Crossref, Web of Science, Google Scholar
C. E. Soteros, D. W. Sumners and S. G. Whittington, J. Knot Theory Ramifications 8, 49 (1999), DOI: 10.1142/S0218216599000067. Link, Web of Science, Google Scholar
D. W. Sumners and S. G. Whittington, J. Phys. A: Math. Gen. 21, 1689 (1988), DOI: 10.1088/0305-4470/21/7/030. Crossref, Google Scholar
S. Suzuki, Kobe J. Math. 1, 19 (1984). Web of Science, Google Scholar