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Abstract: We will discuss a simple combinatorial construction of link and 3-manifold invariants which involves counting trivalent graphs. This construction may be interpreted in terms of counting surfaces of a certain genus and number of boundary components in a Gauss diagram. For 3-manifolds, counting graphs of genus zero and one give the homology and the Casson-Walker invariant respectively.
See also A simple formula for the Casson-Walker invariant (with S. Matveev), J. Knot Theory 18 (2009) 841-864.
