Denoting Regular Knot Projections

Let a knot projection P consisting of N crossings (double points), such as for instance the one viewed here. One converts this projection into a set of pairs of numbers as follows.

First, one chooses a point of the projection which is not a crossing point, and one of the two possible orientations. Then one proceeds along the projection and assigns successive natural numbers to the crossing points, starting from 1 and ending to 2N. At each crossing one assigns two numbers, one for the overcrossing and one for the undercrossing. If for instance one were to start at the left of point A and proceed towards A, point A would be assigned the numbers 1 for overcrossing and 4 for undercrossing; B would be assigned 3 and 6 respectively, C would be assigned 5 and 8 and D 7 and 2.

Once this assignment is completed, one forms a set consisting of N pairs of numbers; each pair corresponds to a crossing, its left element being the overcrossing number, while its right the undercrossing. In the example shown here, the set would be equal to {(1,4), (3,6), (5,8), (7,2)}.

One should notice that the pairs of numbers are ordered, since if one were to exchange their order, an overcrossing would become undercrossing and vice versa. If one were to replace them with unordered pairs, one would obtain the shadow of the knot.

Historically one should mention that this notation is related to the Gauss words , which Gauss used to study closed curves on a plane. What Gauss actually denoted were the knot shadows, and thus in his notation the pairs are unordered. More recently, in 1983 H. Dowker and M. Thisthlethwaite used a similar notation to discuss knot tabulations.

Once now we are given some (regular) knot projection, it is simple to obtain (at least) one such pair of numbers that denotes the projection. A number of questions however arise.

  1. Is the notation unique?

  2. Given a notation, is there always some corresponding knot projection, and is such a projection unique?

  3. What happens to a notation if a Reidemeister move is performed on the corresponding projection?

  4. Is it possible from the notation to apply color tests and/or obtain knot characteristics?

We shall discuss these issues starting from here.

Charilaos Aneziris,

Copyright 1995


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