Partial Petrial polynomials of ribbon graphs equal sums over bouquets, and a signed modification is a 4-invariant for signed simple graphs.
Lando, On a Hopf algebra in graph theory, Journal of Combinatorial Theory, Series B , 80 (1) (2000): 104–121
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Partial Petrial Polynomials of Ribbon Graphs
Partial Petrial polynomials of ribbon graphs equal sums over bouquets, and a signed modification is a 4-invariant for signed simple graphs.