Montesinos Notation

A pretzel knot is formed from a planar unlink of two components by joining the two components with parallel twisted bands. For instance, P(2,5,-7) is formed by adding bands with 2, 5, and -7 half twists.

A Montesinos knot is formed in the same way, except the twists are replaced with rational tangles, and the knot is denoted M( p1/q1 , p2/q2 , .... ). Thus, P(2,5,-7) = M(2,5,-7).

As a basic example, the left handed (2,-5)-torus knot is the Montesinos knot M(1,1,1,1,1) = M(1/5).

Some times the notation isolates a single integer entry, M(n; p1/q1 , p2/q2 , .... ), which is the same knot as M(1/n, p1/q1, p2/q2 ...).

Castellano-Macías and Owad provided the data for 12 crossing knots. The algorithm is described in [3] which contains links to a posting of the data.

The data needs to be checked for orientations.

Specific Knots


[1] Dunfield, N., "A table of boundary slopes of Montesinos knots Topology," 40 (2001), 309-315

[2] Hatcher, A. and Oertel, U., "Boundary slopes for Montesinos knots," Topology 28 (1989), 453-480.

[3] Castellano-Macías, F. and Owad, N., "The tunnel number of all 11 and 12 crossing alternating knots," arXiv preprint.