Algebraic tangles

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

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

1) any $n$-tangle with $0$ or $1$ crossing is $n$-algebraic;

2) if $A$ and $B$ are $n$-algebraic tangles, then $r ^ { i } ( A ) * r ^ { j } ( B )$ is $n$-algebraic for any integers $i$, $j$, where $r$ denotes the rotation of a tangle by the angle $\pi /n$ and $*$ denotes (horizontal) composition of tangles.

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

iii) If an $m$-tangle, $T$, is obtained from an $( n , k )$-algebraic tangle (respectively, an $n$-algebraic tangle) by partially closing $2 n - 2 m$ of its endpoints without a crossing, then $T$ is called an $( n , k )$-algebraic $m$-tangle, respectively an $n$-algebraic $m$-tangle. For $m = 0$ one obtains an $( n , k )$-algebraic link, respectively an $n$-algebraic link.

$2$-algebraic tangles were introduced by J.H. Conway (they are often called algebraic tangles in the sense of Conway or arborescent tangles). The $2$-fold branched covering of $S ^ { 3 }$ with a $2$-algebraic link as a branched set is a Waldhausen graph manifold. Thus, not every link is $2$-algebraic. It is an open problem (as of 2001) to find, for a given $n$, a link which is not $n$-algebraic. The smallest $n$ for which a link $L$ is $n$-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 $8 _ { 18 }$ knot is equal to $3$.

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