Research Projects and Background
The first day of the retreat we will give background talks on topics related to the dimer model, knot invariants, and graph invariants. We will determine at the end of the day which projects to work on, and divide into working subgroups accordingly. Below is a list of some possible projects, and a few sources if you would like to read a little background information before the workshop. None of the readings are required, and we will discuss the background material in the Monday talks and as needed in the working subgroups.
Projects:
Understanding the dimer graph:
• Find an efficient weighting system to encode crossing information that can be used universally
• Translate the Reidemeister moves
• Translate basic graph properties such as vertex size, edge size, connectivity, and width to the language of knot invariants
• Translate smoothings.
Tangles and graph minors:
• Relate subgraphs to tangles
• Dimer graphs as minors of grid graphs
• Describe the dimer graph in terms of Conway's notation for knots. Implement relevant computations in these terms.
Numerical invariants:
• Determine bridge number graphically
• Determine tunnel number graphically
• Prove the Schirmer conjecture about crossing number and bridge number
Twisted Alexander polynomial:
• Study the twisted dimer graph to compare representations
• Study graph theoretic properties of the twisted dimer graph
Kauffman's clock lattice:
• Study the graph of perfect matchings of the dimer graph, which is
Kauffman's clock lattice
• Study the algebra of the clock lattice
Recommended Reading:
Modern Graph Theory, Belá Bollobás
Knots, Gerhard Burde and Heiner Zieschang, available on Google books
A Dimer Model for the Jones Polynomial of Pretzel Knots, M. Cohen, arXiv:1011.3661
A Twisted Dimer Model for Knots, M. Cohen, O. Dasbach, H. Russell, arXiv:1010.5228
Heegaard Splittings of Knot Exteriors, Y. Moriah, arXiv:0608137
Knots and Links, Dale Rolfsen
