1.11.3 Non-invertible arborescent knots

The simplest class of arborescent KLs giving non-invertible knots is (r1,r2) (r3,r4), where ri (i=1,,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
r2, and r3 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,,ri) (r1,r2,,rj). The parts r1,r2,,ri and r1,r2,,rj will be called the pretzel parts of the knot. The term (r1,r2,,ri) (r1,r2,,rj) 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,
,ri) (r1,r2,,rj) 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 rk=rk (k=1,,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.

 

 

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