Algebraic & Geometric Topology

Intrinsic linking and knotting in virtual spatial graphs

Thomas Fleming and Blake Mellor

Full-text: Open access


We introduce a notion of intrinsic linking and knotting for virtual spatial graphs. Our theory gives two filtrations of the set of all graphs, allowing us to measure, in a sense, how intrinsically linked or knotted a graph is; we show that these filtrations are descending and nonterminating. We also provide several examples of intrinsically virtually linked and knotted graphs. As a byproduct, we introduce the virtual unknotting number of a knot, and show that any knot with nontrivial Jones polynomial has virtual unknotting number at least 2.

Article information

Algebr. Geom. Topol., Volume 7, Number 2 (2007), 583-601.

Received: 19 June 2006
Accepted: 13 March 2007
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25]
Secondary: 57M27: Invariants of knots and 3-manifolds

spatial graph intrinsically linked intrinsically knotted virtual knot


Fleming, Thomas; Mellor, Blake. Intrinsic linking and knotting in virtual spatial graphs. Algebr. Geom. Topol. 7 (2007), no. 2, 583--601. doi:10.2140/agt.2007.07.583.

Export citation


  • J H Conway, C M Gordon, Knots and links in spatial graphs, J. Graph Theory 7 (1983) 445–453
  • H A Dye, L H Kauffman, Minimal surface representations of virtual knots and links, Algebr. Geom. Topol. 5 (2005) 509–535
  • E Flapan, J Pommersheim, J Foisy, R Naimi, Intrinsically $n$-linked graphs, J. Knot Theory Ramifications 10 (2001) 1143–1154
  • T Fleming, B Mellor, Virtual Spatial Graphs
  • J Foisy, Intrinsically knotted graphs, J. Graph Theory 39 (2002) 178–187
  • J Foisy, A newly recognized intrinsically knotted graph, J. Graph Theory 43 (2003) 199–209
  • L H Kauffman, Invariants of graphs in three-space, Trans. Amer. Math. Soc. 311 (1989) 697–710
  • L H Kauffman, Virtual knot theory, European J. Combin. 20 (1999) 663–690
  • T Kohara, S Suzuki, Some remarks on knots and links in spatial graphs, from: “Knots 90 (Osaka, 1990)”, de Gruyter, Berlin (1992) 435–445
  • W B R Lickorish, An introduction to knot theory, Graduate Texts in Mathematics 175, Springer, New York (1997)
  • R Motwani, A Raghunathan, H Saran, Constructive Results from Graph Minors: Linkless Embeddings, from: “Proc. of 29th Annual Symposium on Foundations of Computer Science”, IEEE, White Plains, NY (1988) 398–409
  • N Robertson, P Seymour, R Thomas, Sachs' linkless embedding conjecture, J. Combin. Theory Ser. B 64 (1995) 185–227
  • S Yamada, An invariant of spatial graphs, J. Graph Theory 13 (1989) 537–551