3.2.10 KLs and selfavoiding curvesThis 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 This simple idea is a part of his more extended unpublished work.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 2ngon with n pairs of collapsed points. If we put a twosided 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 selfavoiding path, dividing the plane Â^{2}or sphere surface S^{2} into two regions (interior and exterior). On a sphere, those two regions are equivalent. Next figure shows two selfavoiding curves derived from Borromean rings (represented as a Schlegel diagram of an octahedron), and a selfavoiding curve derived from the fullerene C_{60} by the midedge truncation. The number of mirrors of each kind necessary to convert any minimal KL shadow of a given KL into a selfavoiding curve is an invariant of the KL. For both selfavoiding curves derived from Borromean rings, the corresponding number of mirrors is {3,3}. If we denote mirror points of a selfavoiding curve by 1, 2, ..., 2n, we conclude that every selfavoiding curve is an 2ngon 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 selfavoiding 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 selfavoiding 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 selfavoiding curves can be written in a more concise form. First, we can write our code as a sorted list of ordered pairs
