Osaka Journal of Mathematics

Isotopy and homotopy invariants of classical and virtual pseudoknots

François Dorais, Allison Henrich, Slavik Jablan, and Inga Johnson

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Pseudodiagrams are knot or link diagrams where some of the crossing information is missing. Pseudoknots are equivalence classes of pseudodiagrams, where equivalence is generated by a natural set of Reidemeister moves. In this paper, we introduce a Gauss-diagrammatic theory for pseudoknots which gives rise to the notion of a virtual pseudoknot. We provide new, easily computable isotopy and homotopy invariants for classical and virtual pseudodiagrams. We also give tables of unknotting numbers for homotopically trivial pseudoknots and homotopy classes of homotopically nontrivial pseudoknots. Since pseudoknots are closely related to singular knots, this work also has implications for the classification of classical and virtual singular knots.

Article information

Osaka J. Math. Volume 52, Number 2 (2015), 409-423.

First available in Project Euclid: 24 March 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds


Dorais, François; Henrich, Allison; Jablan, Slavik; Johnson, Inga. Isotopy and homotopy invariants of classical and virtual pseudoknots. Osaka J. Math. 52 (2015), no. 2, 409--423.

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  • C.C. Adams: The Knot Book, Freeman, New York, 1994.
  • R. Hanaki: Pseudo diagrams of knots, links and spatial graphs, Osaka J. Math. 47 (2010), 863–883.
  • A. Henrich, R. Hoberg, S. Jablan, L. Johnson, E. Minten and L. Radović: The theory of pseudoknots, J. Knot Theory Ramifications 22 (2013), 1350032.
  • A. Henrich and S. Jablan: On the coloring of pseudoknots, arXiv:1305.6596 (2013).
  • A. Henrich, N. MacNaughton, S. Narayan, O. Pechenik and J. Townsend: Classical and virtual pseudodiagram theory and new bounds on unknotting numbers and genus, J. Knot Theory Ramifications 20 (2011), 625–650. \interlinepenalty10000
  • S. Jablan and R. Sazdanović: LinKnot, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007,
  • L.H. Kauffman: Virtual knot theory, European J. Combin. 20 (1999), 663–690.
  • M. Polyak: Minimal generating sets of Reidemeister moves, Quantum Topol. 1 (2010), 399–411.
  • Y. Nakanishi: Unknotting numbers and knot diagrams with the minimum crossings, Math. Sem. Notes Kobe Univ. 11 (1983), 257–258.
  • S.A. Bleiler: A note on unknotting number, Math. Proc. Cambridge Philos. Soc. 96 (1984), 469–471.
  • S. Jablan and R. Sazdanović: Unlinking number and unlinking gap, J. Knot Theory Ramifications 16 (2007), 1331–1355.v1.
  • J.A. Bernhard: Unknotting numbers and minimal knot diagrams, J. Knot Theory Ramifications 3 (1994), 1–5.