1.11 Non-invertible KLs

An orientation of a knot is defined by choosing a direction to travel around the knot. Hence, for every non-oriented knot K, we have two different oriented knots denoted by K and K". A knot K is called invertible if the oriented knots K and K" are equivalent. Otherwise, it is called non-invertible. The existence of non-invertible knots was shown by H.F. Trotter (1963), who discovered the non-invertible knot 7,5,3 and the whole family of non-invertible pretzel knots (2p+1),(2q+1),(2r+1) (p q, p r, q r). Unlike in 1963, today we know that almost all knots are non-invertible (Murasugi, 1996). The number of non-invertible and the number of invertible knots with 3 n 16 crossings is given in the following table (the sequences A052403 and A052402 from the The On-Line Encyclopedia of Integer Sequences by N. Sloane):

0 0 0 0 0 1 2 33 187 1144 6919 38118 226581 1309875
1 1 2 3 7 20 47 132 365 1032 3069 8854 26712 78830

The first non-invertible knot from Rolfsen's knot tables is the knot 817 (.2.2 in the Conway notation). By composing two copies of this knot, one with the matching orientations, and the other with different orientations, we get two distinct composite knots which are not equivalent. A. Kawauchi (1979) proved that there is no deformation of the knot .2.2 that reverses the orientation of the knot. The thirty six non-invertible knots with ten crossings of fewer are identified by R. Hartley (1983). So far, no one has developed a general technique to recognize non-invertible knots.

In recognizing achiral knots, we can use their antisymmetrical rigid representations in 3, or on the sphere S3. Every KL that has an antisymmetrical presentation in 3, has it on S3 as well, but not necessarily vice versa. For example, figure-eight knot has both presentations antisymmetrical: its non-minimal diagram is invariant with regard to a rotational antireflection of order 4 (i.e., rotational reflection followed by vertex sign change), and its diagram coming from S3 is centro-antisymmetrical. Achiral knot 817 has a more remarkable property: it has a centro-antisymmetrical presentation coming from S3, but has no antisymmetrical presentation in 3. Therefore, it is a topological rubber glove in 3 (Flapan, 1998, 2000).

The program Snappea by J. Weeks computes knot symmetry group and recognizes non-invertible knots. For checking invertibility of knots you can use also the program Knotscape containing Snappea 2.0 as its part.