Abstract:
Knotted circles and the space around them carry a surprising wealth of mathematical data, including connections to the quantum and relativistic worlds.  I will describe how certain surfaces called checkerboards help parse this data, paying special attention to a gluing operation called plumbing. We will see that, under plumbing, some classes of checkerboards have unique factorizations, and others do not. Finally, with a focus on alternating knots, I will connect these ideas to the notions of representativity and Khovanov homology.