Osaka Journal of Mathematics

Pseudo diagrams of knots, links and spatial graphs

Ryo Hanaki

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A pseudo diagram of a spatial graph is a spatial graph projection on the $2$-sphere with over/under information at some of the double points. We introduce the trivializing (resp. knotting) number of a spatial graph projection by using its pseudo diagrams as the minimum number of the crossings whose over/under information lead the triviality (resp. nontriviality) of the spatial graph. We determine the set of non-negative integers which can be realized by the trivializing (resp. knotting) numbers of knot and link projections, and characterize the projections which have a specific value of the trivializing (resp. knotting) number.

Article information

Osaka J. Math., Volume 47, Number 3 (2010), 863-883.

First available in Project Euclid: 24 September 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M15: Relations with graph theory [See also 05Cxx]


Hanaki, Ryo. Pseudo diagrams of knots, links and spatial graphs. Osaka J. Math. 47 (2010), no. 3, 863--883.

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  • N. Askitas and E. Kalfagianni: On knot adjacency, Topology Appl. 126 (2002), 63--81.
  • D. Bar-Natan: On the Vassiliev knot invariants, Topology 34 (1995), 423--472.
  • J.S. Birman and X.-S. Lin: Knot polynomials and Vassiliev's invariants, Invent. Math. 111 (1993), 225--270.
  • A.D. Bates and A. Maxwell: DNA Topology, 2nd ed., Oxford university press, 2005.
  • T.D. Cochran and R.E. Gompf: Applications of Donaldson's theorems to classical knot concordance, homology $3$-spheres and property $P$, Topology 27 (1988), 495--512.
  • P.R. Cromwell: Homogeneous links, J. London Math. Soc. (2) 39 (1989), 535--552.
  • F.B. Dean, A. Stasiak, Th. Koller, N.R. Cozzarelli: Duplex DNA knots produced by Escherichia coli topoisomerase, I, structure and requirements for formation, J. Biol. Chem. 260 (1985), 4975--4983.
  • C.H. Dowker and M.B. Thistlethwaite: Classification of knot projections, Topology Appl. 16 (1983), 19--31.
  • Y. Huh and K. Taniyama: Identifiable projections of spatial graphs, J. Knot Theory Ramifications 13 (2004), 991--998.
  • L.H. Kauffman: Invariants of graphs in three-space, Trans. Amer. Math. Soc. 311 (1989), 697--710.
  • W.K. Mason: Homeomorphic continuous curves in $2$-space are isotopic in $3$-space, Trans. Amer. Math. Soc. 142 (1969), 269--290.
  • K. Murasugi: Knot Theory & its Applications, Birkhäuser Boston, Boston, MA, 2008.
  • R. Nikkuni: A remark on the identifiable projections of planar graphs, Kobe J. Math. 22 (2005), 65--70.
  • R. Nikkuni: Completely distinguishable projections of spatial graphs, J. Knot Theory Ramifications 15 (2006), 11--19.
  • J.H. Przytycki: Positive knots have negative signature, Bull. Polish Acad. Sci. Math. 37 (1989), 559--562 (1990).
  • J.H. Przytycki and K. Taniyama: Almost positive links have negative signature, J. Knot Theory Ramifications 19 (2010), 187--289.
  • T. Stanford: Finite-type invariants of knots, links, and graphs, Topology 35 (1996), 1027--1050.
  • A. Stoimenow: Gauss diagram sums on almost positive knots, Compos. Math. 140 (2004), 228--254.
  • S. Suzuki: Local knots of $2$-spheres in $4$-manifolds, Proc. Japan Acad. 45 (1969), 34--38.
  • K. Taniyama: A partial order of knots, Tokyo J. Math. 12 (1989), 205--229.
  • K. Taniyama: A partial order of links, Tokyo J. Math. 12 (1989), 475--484.
  • K. Taniyama: Knotted projections of planar graphs, Proc. Amer. Math. Soc. 123 (1995), 3575--3579.
  • K. Taniyama and C. Yoshioka: Regular projections of knotted handcuff graphs, J. Knot Theory Ramifications 7 (1998), 509--517.
  • K. Taniyama and A. Yasuhara: Clasp-pass moves on knots, links and spatial graphs, Topology Appl. 122 (2002), 501--529.
  • K. Taniyama: Unknotting numbers of diagrams of a given nontrivial knot are unbounded, J. Knot Theory Ramifications 18 (2009), 1049--1063.
  • P. Traczyk: Nontrivial negative links have positive signature, Manuscripta Math. 61 (1988), 279--284.
  • V.A. Vassiliev: Cohomology of knot spaces; in Theory of Singularities and its Applications, Adv. Soviet Math. 1, Amer. Math. Soc., Providence, RI, 23--69, 1990.
  • S. Yamada: An invariant of spatial graphs, J. Graph Theory 13 (1989), 537--551.
  • M. Yamamoto: Knots in spatial embeddings of the complete graph on four vertices, Topology Appl. 36 (1990), 291--298.