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3. Linear world

The origin (or O-world) is the basic polyhedron 1 - 4-regular graph with one vertex, the usual simbol of infinity (¥).  The first, linear world (or L-world) contains only one basic link: a digon 2. From it, we derive the infinite family p, consisting of p-gons with digonal edges (p > 2). For p odd we have the infinite series of knots 3, 5, 7, ¼ (or 31, 51, 71,¼), and for p even the infinite series of 2-component links 2, 4, 6,¼ (or 212, 412, 612,¼). For each of them, the Alexander polynomial D(t) for knots, or reduced Alexander polynomial D(t,t) for 2-component links, is an alternating polynomial of order p+1, with all coefficients equal to 1. For every knot p of this family, the unknoting number is u(p) = (p-1)/2. All knots of L-world are periodical knots with graph symmetry group G = [2,p], and with knot symmetry group G' = [2,p]+, generated by p-rotation and 2-rotation. The periods of every such knot are p and 2 [9].

 

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