1.13.1 Applications of minimum braids and braid family representatives
A graph-theoretical approach to knot theory is proposed by A. Caudron (1982). T. Gittings (2004, Conjecture 1) established a mapping between minimum braids with s strands and trees with s+1 vertices and conjectured that the number of graph trees of n vertices with alternating minimum braids is equal to the number of rational KLs with n crossings.
KL is achiral (or amphicheiral) if its "left" and "right" forms are equivalent, meaning that one can be transformed to the other by an ambient isotopy. If an oriented knot or link L can be represented by an antisymmetrical vertex-bicolored graph on a sphere, whose vertices with the sign +1 are white, and vertices with the sign -1 are black, it is achiral. In this case, for an oriented knot or link L there exists an antisymmetry (sign-changing symmetry) switching orientations of vertices, i.e., mutually exchanging vertices with the signs +1 and -1. In the language of braid words, this means that its corresponding braid word is antisymmetric (or palindromic): there exist a mirror antisymmetry transforming one letter to another and vice versa and changing their case (i.e., transforming capital to lower case letters and vice versa). For example, the reduced braid words Ab |Ab or ABac |BDcd are palindromic, where the anti-mirror is denoted by |. Hence, we believe that the origin of all achiral oriented KLs are palindromic reduced braids.
Conjecture An oriented KL is achiral iff it can be obtained from a palindromic reduced braid by a symmetric assigning of degrees.
For s = 2 all alternating
BFRs are of the form (Ab)n ( n ³
2), defining a series of
the basic polyhedra (2n)*, beginning with 2 2, .1 = 6*,
8*, 10*, 12*, etc. All of them
are achiral KLs, representing a source of other achiral KLs. From 4:1-01
AbAb (2 2 or 41) by a symmetric assigning of degrees we can
derive achiral alternating knots with n £
10 crossings: 6:1-02 A2bAb2
(2 1 1 2 or 63), 8:1-05 A3bAb3 (3 1 1
3 or 89), 10:1-017 A3b2A2b3
((3,2) (3,2) or 1079), and one achiral alternating link with
crossings: 8:3-05a A2b2A2b2
((2,2) (2,2) or 843), etc. In general, from
AbAb the following families of achiral alternating KLs are derived: