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Gauss Code

A Gauss code for a virtual knot diagram can be obtained by numbering all of the non-virtual crossings, then beginning at some point and travelling along the diagram. At each crossing, indicate whether you are currently on the over or on the under strand, and whether this is a positive or a negative crossing. Virtual crossings do not show up in the Gauss code. For example, U3+ indicates the under strand of crossing number 3, which is a positive crossing.

The gauss code of virtual knot 3.5, shown below, read beginning at the arrow, is O1-O2-O3-U1-U2-U3-.

Behaviour on Mirror Images and Inverses

A Gauss code for the vertical mirror image of a virtual knot diagram can be obtained from the original Gauss code by replacing every O with U and vice-versa, as well as every + with -, and vice-versa. Taking the horizontal mirror image replaces + with - and vice-versa. A Gauss code for an inverse is simply the original Gauss code backwards.