We owe the T-shirt design and our 2008 contest logo to professor Neal W. Stoltzfus of LSU Math Department.

Quasitree

In the study of connected ribbon graphs (that is, connected graphs, together with an embedding in an oriented surface with polygonal faces), a quasi-tree is a sub-ribbon graph which contains a spanning tree and sufficient additional edges to "minimally carry" the quasi-tree's homology.

Recall that a connected graph with v vertices and e edges has a spanning tree with the same vertex set and only v - 1 edges. Similarly, a quasi-tree of a ribbon graph is a connected, spanning sub-ribbon graph with only one face and e = v+2 g - 1 edges, for an integer g called the genus.

This concept was used by in a paper "Quasi-tree expansion for the Bollobás-Riordan-Tutte polynomial" by Oliver T. Dasbach, David Futer, Efstratia Kalfagianni, Xiao-Song Lin and Neal W.
Stoltzfus to prove that the determinant of a knot is the alternating sum (with respect to genus) of
the number of quasitrees in a certain ribbon graph constructed from a knot projection.

For more information and applications, see
http://www.math.lsu.edu/~stoltz/quasitree.pdf