Question

Hey guys,

The following paper uses the term `bridge' in their definition of the Tutte polynomial:

Bennett Thompson, David J. Pearce, Craig Anslow, and Gary Haggard. Visualizing the computation tree of the tutte polynomial. In Proceedings of the 4th ACM sympo- sium on Software visualization, SoftVis ’08, pages 211–212, New York, NY, USA, 2008. ACM. Available from: http://doi.acm.org/10.1145/1409720.1409760, doi:http: //doi.acm.org/10.1145/1409720.1409760.

However, the Wiki page and other papers use the term `crossing'.

Are these the same thing or am I confusing them? What do you think?

Thank you.

Answer 1

The Wikipedia page for the Tutte polynmomial doesn't use the word crossing, it also uses the word bridge. In graph theory, a bridge of a connected graph is an edge that separates the graph into two components.

However, there is a relation between the Tutte polynomial and the Jones and HOMFLY polynomials. More precisely, the HOMFLY polynomial generalizes the Tutte polynomial for planar graph. A knot diagram has crossings, which means points where two arcs of the knot cross. A knot diagram also has bridges; a bridge is a maximal sequence of over-crossings along an arc of the diagram. So there is a little bit of collision of terminology, because crossings aren't bridges and because bridges for knots aren't the same as bridges for graphs.