3.2.10  KLs and self-avoiding curves

This part of the work is inspired by a series of sculptures titled Viae Globi, created by Carlo Sequin (2001), and by a conversation with Haresh Lalvani, who proposed to identify vertices of a polygon, in particular two vertices of a triangle in order to obtain a "triangle with two vertices". This simple idea is a part of his more extended unpublished work. Generalizing this idea, we can conclude that every KL shadow with n crossings is an 2n-gon with n pairs of collapsed points.

If we put a two-sided mirror in every vertex of a KL shadow, by a suitable choice of mirror positions, as the result we can obtain a single closed mirror curve. The choice of mirror positions is made according to the rules for obtaining a single mirror curve. This kind of mirror curve is a self-avoiding path, dividing the plane Â2or sphere surface S2 into two regions (interior and exterior). On a sphere, those two regions are equivalent. Next figure shows two self-avoiding curves derived from Borromean rings (represented as a Schlegel diagram of an octahedron), and a self-avoiding curve derived from the fullerene C60 by the mid-edge truncation. The number of mirrors of each kind necessary to convert any minimal KL shadow of a given KL into a self-avoiding curve is an invariant of the KL. For both self-avoiding curves derived from Borromean rings, the corresponding number of mirrors is {3,3}. 

If we denote mirror points of a self-avoiding curve by 1, 2, ..., 2n, we conclude that every self-avoiding curve is an 2n-gon with n pairs of identified points. In order to avoid loops, we never identify adjacent points. If we denote points belonging to internal mirrors by underlined numbers, and overline numbers belonging to external mirrors, the first self-avoiding curve can be denoted by the code 

Analogously to Gauss codes, this code will depend on a beginning point and orientation. Hence, several codes can be assigned to every self-avoiding curve and we can choose the minimal one as a representative. If we think of the exterior and interior of a curve as equivalent, then a code and its dual (the code with inverted underlinings and overlinings) are considered to be the same. In the same way as with Gauss and Dowker codes, the proposed codes for self-avoiding curves can be written in a more concise form. First, we can write our code as a sorted list of ordered pairs