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).
Let be given an oriented knot projection D with generators g_{1},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_{n}t^{n}+¼+c_{1}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_{n}t^{n}+¼+c_{k}t^{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) = x^{2}y^{2}z^{2}, 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. |