1.11.2 Noninvertible pretzel knots Every alternating pretzel KL is of the form r_{1},r_{2},¼,r_{i}, where r_{i} are Rtangles (i ³ 3). Pretzel knots are obtained as the numerator closures of the pretzel tangles of the reduced type [1], or [¥]. Hence, nonreduced types of pretzel KLs can be:
The main question is wether it is possible to determine
noninvertibility of pretzel knots according to tangle types.
Every pretzel knot can be drawn as a regular tgon with vertices
denoting Rtangles, called tdiagram. In a tdiagram
vertices by themselves are treated as symmetrical, and the mirror
line contains at least one vertex.
Conjecture (Noninvertibility criterion for pretzel knots) A pretzel knot is noninvertible iff its type symbol consists only from 0s and 1s, and its tdiagram is not mirrorsymmetrical. We give detailed description of pretzel knots with n £ 5:
For example, the pretzel knots 3,3,3,5,7 and 3,3,5,3,7 are
noninvertible, while 3,5,5,3,7 is invertible, and all have the
same type [1,1,1,1,1].
In general, the necessary condition for invertibility of pretzel
knots is that the sum of numbers in knot type must be odd.
Sufficiency is determined, based on symmetry condition, from the
tdiagram.
