14. Projections

Treating a tangle T as a portion of a link surrounded by a Conway sphere, a flype is the 2-rotation of T with the "horizontal" rotation axis, expressed by the relationship T+1=1+T_h [14]. According to the Tait Flyping Conjecture (proved in [45]), all reduced projections of an alternating link could be obtained by doing all the possible flypes on any one such projection, without ever changing the number of crossings. Because we are dealing only with reduced alternating projections, the term "projection" will be used in that restricted sense. Every Conway symbol uniquely defines the canonical projection of a link (i.e. the corresponding graph). As before, we will consider only projections with singular digons. All the other projections could be obtained from them replacing singular digons by the chains of digons. All the projections of links with singular digons for n £ 9 are illustrated. They are derived from only several graphs by introducing digons in their vertices (i.e. by the vertex bicoloring).

Every flype changes or not the number of digons. Hence, among all the projections of a link we could distinguish its canonical projection described by the Conway symbol, projections equivalent to the canonical with the same number of digons, and other nonequivalent projections. This could be also expressed by Conway notation (e.g. 222' = 2+,2+, 232' = 2++,2+, 2,2,2++' = 2,2+,2+, 2212' = 21+,2+, etc.). When replacing singular digons by the chains of digons, we could use the canonical projection or any projection equivalent to it, but not the nonequivalent ones, because this will result in an uncomplete derivation. For example, by such replacement from 222 we obtain 322 and 232, and from 222' = 2+,2+ only 3+,2+ = 322. Let us notice that to every source link of P-world corresponds exactly one projection.  

Let be given an oriented knot projection D with generators g₁,¼,g_n. In every vertex of D there is an overcrossing generator g_i, and generators g_j, g_k ending and starting in the vertex. If e is the sign of the vertex, then a_ii = et, a_ij = 1, a_ik = -1, and d(t) = det(a_ij). Two oriented knot projections D' and D'' are nonisomorphic if d'(t) ¹ d''(t) [46].

There are some important properties of the polynomial d(t) = c_ntⁿ+¼+c₁t:  
           (a) for every knot projection, the degree of d(t) is n, |c_n| = 1 and |c₁| = |w(D)|, where w(D) is the writhe of D;  
           (b) d(t) and d(-t) correspond to the obverse (enantiomorphic) knot diagrams;  
           (c) for n = 0(mod 2), a change of the orientation results in the change of d(t) to d(-t), and for n = 1(mod 2) in the change of d(t) to -d(-t).   

According to (b) and (c), in the set of all polynomials d(t) we may distinguish even polynomials (d(t) = d(-t)), containing only even degrees of t, corresponding to amphicheiral knot projections, and odd polynomials (d(t) = -d(-t)), containing only odd degrees of t, which are invariant to a change of orientation. Using this polynomial, we derived the list of all knot projections given in Dowker notation for n £ 12 and identified the amphicheiral projections.  

This polynomial could be directly transferred to link projections. In this case, the result is a polynomial of the form: d(t) = c_ntⁿ+¼+c_kt^k, where n is the number of crossing points, and k is the number of link components. For every link, |c_n| = 1. If a_i are the link components, a_ii = w(a_i), and if a_ij = lk(a_i,a_j) denotes the linking number of the components a_i, a_j, then |c_k| = |det(a_ij)|. In order to increase the selectivity of this polynomial, it is possible to denote generators belonging to different link components by different variables. For example, for Borromean rings d(x,y,z) = x²y²z², so from this we conclude that the components are interchangeable, and that Borromean rings are amphicheiral.  

A similar polynomial, somewhat earlier introduced by C. Liang and Y. Jiang [28], is very efficiently used in [22] for recognizing amphicheiral knots. As it is noticed in [22], the amphicheirality is connected with antisymmetry, where a rotational antireflection produces invertible amphicheiral knots, and a sole 2-antirotation non-invertible amphicheiral knots. We will consider amphicheiral knots as the members of infinite families.