Gauss code

A Gauss code records the successive crossings encountered when traversing an oriented knot diagram. Each crossing label occurs twice; overpassing/underpassing data and crossing signs make the code sufficient to recover its Wirtinger presentation and smoothing counts. Labels are arbitrary and cyclic starting points represent the same traversal.

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.