Basic graph theory

1.11.3 Non-invertible arborescent knots

The simplest class of arborescent KLs giving non-invertible knots is (r₁,r₂) (r₃,r₄), where r_i (i=1,¼,4) are R-tangles. By using tangle type calculation, we conclude that knots will be obtained for the following (r₁,r₂) (r₃,r₄) pretzel type sets: [1,1] [1,1], [0,1] [1,1], [0,0] [1,1], [1,¥] [1,¥], [0,0] [0,1], [0,1] [0,¥], [0,¥] [1,¥], [0,0] [0,0], [0,¥] [0,¥], 2-component links will be obtained for [1,1] [1,¥], [0,1] [0,1], [0,0] [1,¥], [0,1] [¥,¥], [1,¥] [¥,¥], [0,0] [0,¥], [0,¥] [¥,¥], and 3-component links for [1,1] [¥,¥], [0,0] [¥,¥], [¥,¥] [¥,¥].

Conjecture A knot of the form (r₁,r₂)(r₃,r₄) is non-invertible iff r₁ ¹ r₂, and r₃ ¹ r₄; it is achiral non-invertible iff r₁=r₃ and r₂=r₄, and chiral non-invertible otherwise.

For n=10 there are three non-invertible knots of this form. Two of them are achiral: (3,2) (3,2) and (2 1,2) (2 1,2), and one is chiral: (3,2) (2 1,2).

For n=11 there are six of them:

(2 1 1,2) (2 1,2)	(2 1 1,2) (3,2)	(2 2,2) (2 1,2)
(3,2 1) (2 1,2)	(2 2,2) (3,2)	(3,2 1) (3,2)

It is clear that achiral non-invertible knots can be obtained only for (r₁,r₂) (r₃,r₄) knots with the type sets [1,1] [1,1], [1,¥] [1,¥], [0,0] [0,0], [0,¥] [0,¥]. The first achiral non-invertible representatives of those types are: (3,2 1 1) (3,2 1 1), (3,2) (3,2), (2 2,2 1) (2 2,2 1), (2 1,2) (2 1,2), respectively.

Now we will consider a more general case: non-invertible knots of the form (r₁,r₂,¼,r_i) (r₁¢,r₂¢,¼,r_j¢). The parts r₁,r₂,¼,r_i and r₁¢,r₂¢,¼,r_j¢ will be called the pretzel parts of the knot. The term (r₁,r₂,¼,r_i) (r₁¢,r₂¢,¼,r_j¢) is a knot iff the types of the pretzel parts are [0] [0], [0] [1], [1] [0], [1] [¥], [¥] [1], [¥] [¥]. This means that every pretzel part may have at most one R-tangle of the type [¥]. The pretzel parts are treated as ordered sequences of R-tangles, and not as cyclic structures, as in the case of pretzel KLs. From the symmetry reasons, it is sufficient to consider only knots of the type [0] [0], [0] [1], [1] ¥], ¥] [¥].

Conjecture A knot of the form (r₁,r₂,¼,r_i) (r₁¢,r₂¢,¼,r_j¢) of the type [0] [0], [1] [¥], [¥] [1], [¥] [¥] is non-invertible iff none of its pretzel parts is mirror-symmetrical; it is chiral non-invertible iff i=j and r_k=r_k¢ (k=1,¼,i), and chiral non-invertible otherwise. A knot of the type [0] [1], [1] [0] is chiral non-invertible iff its pretzel part of the type [0] is not mirror-symmetrical.