Partial Petrial polynomials of ribbon graphs equal sums over bouquets, and a signed modification is a 4-invariant for signed simple graphs.
Chmutov, Partial-dual genus polynomial as a weight system, Communications in Mathe- matics, 31 (2023): 113-124
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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.