1.11.3 Noninvertible
arborescent knots The simplest class of arborescent KLs giving noninvertible knots is (r_{1},r_{2}) (r_{3},r_{4}), where r_{i} (i=1,¼,4) are Rtangles. By using tangle type calculation, we conclude that knots will be obtained for the following (r_{1},r_{2}) (r_{3},r_{4}) 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,¥], 2component links will be obtained for [1,1] [1,¥], [0,1] [0,1], [0,0] [1,¥], [0,1] [¥,¥], [1,¥] [¥,¥], [0,0] [0,¥], [0,¥] [¥,¥], and 3component links for [1,1] [¥,¥], [0,0] [¥,¥], [¥,¥] [¥,¥]. Conjecture A knot of the form (r_{1},r_{2})(r_{3},r_{4}) is noninvertible iff r_{1} ¹ r_{2}, and r_{3} ¹ r_{4}; it is achiral noninvertible iff r_{1}=r_{3} and r_{2}=r_{4}, and chiral noninvertible otherwise. For n=10 there are three noninvertible 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:
It is clear that achiral noninvertible knots can be obtained only for (r_{1},r_{2}) (r_{3},r_{4}) knots with the type sets [1,1] [1,1], [1,¥] [1,¥], [0,0] [0,0], [0,¥] [0,¥]. The first achiral noninvertible 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: noninvertible knots of the form (r_{1},r_{2},¼,r_{i}) (r_{1}¢,r_{2}¢,¼,r_{j}¢). The parts r_{1},r_{2},¼,r_{i} and r_{1}¢,r_{2}¢,¼,r_{j}¢ will be called the pretzel parts of the knot. The term (r_{1},r_{2},¼,r_{i}) (r_{1}¢,r_{2}¢,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 Rtangle of the type [¥]. The pretzel parts are treated as ordered sequences of Rtangles, 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 (r_{1},r_{2},¼,r_{i}) (r_{1}¢,r_{2}¢,¼,r_{j}¢) of the type [0] [0], [1] [¥], [¥] [1], [¥] [¥] is noninvertible iff none of its pretzel parts is mirrorsymmetrical; it is chiral noninvertible iff i=j and r_{k}=r_{k}¢ (k=1,¼,i), and chiral noninvertible otherwise. A knot of the type [0] [1], [1] [0] is chiral noninvertible iff its pretzel part of the type [0] is not mirrorsymmetrical.
