Classification of knot projections

Knots of genus one or on the number of alternating knots of given genus

“…It is easily seen that for p, q, r even D is not realizable (or it can be formally checked using the conditions in [10]) hence p, q, r are odd and D is the pretzel diagram P (p, q, r).…”

A prime curve is determined by its $VF$-matrix

“…For example, the Gauss code of the curve depicted in Figure 1 is 1, 2, 3, 1, 4, 3, 2, 5, 5, 4. (See [1] for a description of Dowker notation, which is closely related and more compact.) In this section we present an algorithm that extracts the Gauss code of a prime curve directly from the curve's VF -matrix.…”

“…We assume that S 1 and S 2 are oriented, and that a basepoint * has been chosen for S 1 . By generic, we mean that the following conditions are satisfied:…”

“…Note that k( * ) is not a double point. To each double point, assign a standard basis vector e i of (Z/2Z) n as follows: Start at k( * ) and trace along k(S 1 ) in the positive direction, as determined by the orientation on S 1 . To the first double point encountered, assign the vector e 1 .…”

Classifying Immersed Curves

“…He mused about when such a sequence would give the projection of a knot. This problem was solved by Treybig [15], Marx [12], Lovasz and Marx [11], Read and Rosenstiehl [14], and Dowker and Thistlethwaite [7]. In these works necessary and sufficient criteria were established that determine the "planarity" of a Gauss word.…”