Algebraic tangles

A family of tangles (cf. Tangle) defined recursively for any as follows:

i) -algebraic tangles is the smallest family of -tangles satisfying

1) any -tangle with or crossing is -algebraic;

2) if and are -algebraic tangles, then is -algebraic for any integers , , where denotes the rotation of a tangle by the angle and denotes (horizontal) composition of tangles.

ii) If in condition 2) above, is restricted to tangles with no more than crossings, one obtains the family of -algebraic tangles.

iii) If an -tangle, , is obtained from an -algebraic tangle (respectively, an -algebraic tangle) by partially closing of its endpoints without a crossing, then is called an -algebraic -tangle, respectively an -algebraic -tangle. For one obtains an -algebraic link, respectively an -algebraic link.

-algebraic tangles were introduced by J.H. Conway (they are often called algebraic tangles in the sense of Conway or arborescent tangles). The -fold branched covering of with a -algebraic link as a branched set is a Waldhausen graph manifold. Thus, not every link is -algebraic. It is an open problem (as of 2001) to find, for a given , a link which is not -algebraic. The smallest for which a link is -algebraic is called the algebraic index of the link (it is bounded from above by the braid and bridge indices of the link). For example, the algebraic index of the knot is equal to .

References

[a1]	J.H. Conway, "An enumeration of knots and links" J. Leech (ed.) , Computational Problems in Abstract Algebra , Pergamon Press (1969) pp. 329–358
[a2]	J.H. Przytycki, T. Tsukamoto, "The fourth skein module and the Montesinos–Nakanishi conjecture for -algebraic links" J. Knot Th. Ramifications (2001)