We owe the Tshirt 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 quasitree is a subribbon graph
which contains a spanning tree and sufficient additional
edges to "minimally carry" the quasitree'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 quasitree of a ribbon graph
is a connected, spanning subribbon 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 "Quasitree
expansion for the BollobásRiordanTutte polynomial"
by Oliver T. Dasbach, David Futer, Efstratia Kalfagianni,
XiaoSong 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
