Framed $4$-graphs: Euler tours, Gauss circuits and rotating circuits

We consider connected finite $4$-valent graphs with the structure of opposite edges at each vertex (framed $4$-graphs). For any of such graphs there exist Euler tours, in travelling along which at each vertex we turn from an edge to a nonopposite one (rotating circuits); and at the same time, it is not true that for any such graph there exists an Euler tour passing from an edge to the opposite one at each vertex (a Gauss circuit). The main result of the work is an explicit formula connecting the adjacency matrices of the Gauss circuit and an arbitrary Euler tour. This formula immediately gives us a criterion for the existence of a Gauss circuit on a given framed $4$-graph. It turns out that the results are also valid for all symmetric matrices (not just for matrices realisable by a chord diagram).
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