A Gauss code for a virtual knot diagram is a sequence of crossing labels, each repeated twice and assigned a + or - symbol to identify over and undercrossings. Eliminate these symbols and what remains is a Gauss code for the shadow of the diagram, one type of virtual pseudodiagram. While it is now impossible to determine which particular virtual diagram the shadow resulted from, we can consider the collection of all diagrams, called resolutions of the shadow, that would yield such a code. We compute the average virtual bridge number over all these diagrams and show that for a shadow with n classical precrossings, the average virtual bridge number is .

Keywords:

AMSC: 57M25, 57M27

References

E. Byberi and V. Chernov, Commun. Contemp. Math. 10(), 1013 (2008), DOI: 10.1142/S0219199708003137. Link, Google Scholar
V. Chernov, J. Knot Theory Ramifications 22, 6 (2013), DOI: 10.1142/S0218216512501398. Google Scholar
J. Chung and X. S. Lin, Math. Proc. Cambridge Philos. Soc. 137, 617 (2004), DOI: 10.1017/S0305004104007753. Crossref, Web of Science, Google Scholar
R. Hanaki, Osaka J. Math. 47, 863 (2010). Web of Science, Google Scholar
A. Henrichet al., J. Knot Theory Ramifications 22, 21 (2013), DOI: 10.1142/S0218216513500326. Google Scholar
A. Henrichet al., J. Knot Theory Ramifications 20, 625 (2011), DOI: 10.1142/S0218216511009388. Link, Web of Science, Google Scholar
L. Kauffman, European. J. Combin. 20, 663 (1999), DOI: 10.1006/eujc.1999.0314. Crossref, Web of Science, Google Scholar