= Oriented smoothing permutation of a Gauss code
{title2=$p(j)=\tau(j)+1\pmod{2n}$}
Number the $2n$ visits along an oriented <knot diagram>, and let $\tau$ exchange the two visits to each crossing. At an <oriented smoothing>, an incoming strand follows the outgoing strand of the other visit. Thus the cycles of the displayed <permutation> are exactly the <Seifert circles>. Their count $s$ gives the <Seifert algorithm> genus $(1-s+n)/2$ without requiring a separate drawing of the smoothed diagram.
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