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I cannot find any answer to that apparently simple problem : On a square lattice, a path is given by a sequence of relative moves in {"move forward", "turn right" and "turn left"}.

Is there a rule that characterizes if a path is self-avoiding (or not) ?

Edit : Let me precise the kind of rule I am looking for: Let's imagine a walk described, this time, by absolute moves (U=up, D=downN=move North, R=RightS=move South, L=Left)E=move East, W=move West), then the presence of a loop in the sequence is characterized by a subsequence for which nb(Unb(N) = nb(Dnb(S) and nb(R)=nb(L)nb(E)=nb(W). That's a simple rule.

Is there such a rule in the case of a sequence of relative moves ? Or do we have to translate the sequence to absolute moves ? Thanks.

Example (to make myself clear): here is a walk (or part of a walk), written in absolute moves {North, East, West, South}: EENWNNWSSS => We immediately know it is a loop, without having to draw anything or keep track of the positions visited, because nb(N)=nb(S) and nb(E)=nb(W).

Now here is the same walk written in relative moves {Forward, Turn right, Turn left}: FFLFLFRFFLFLFFF => Without drawing anything, nor converting to absolute moves. Is there a rule that allows to say it is a loop ?

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I cannot find any answer to that apparently simple problem : On a square lattice, a path is given by a sequence of relative moves in {"move forward", "move to the turn right" and "move to the turn left"}.

Is there a rule that characterizes if a path is self-avoiding (or not).not) ?

Edit : Let me precise the kind of rule I am looking for: Let's imagine a walk described, this time, by absolute moves (U=up, D=down, R=Right, L=Left), then the presence of a loop in the sequence is characterized by a subsequence for which nb(U) = nb(D) and nb(R)=nb(L). That's a simple rule.

Is there such a rule in the case of a sequence of relative moves ? Or do we have to translate the sequence to absolute moves ? Thanks.

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