## Arborescent knots

Introduced by J.Conway,
further studied by A.Caudron
and F.Bonahon & L.Siebenmann; used to order the 10-intersection knots in
Rolfsen table.

Also known as double fat.

Distinguished by the possibility to express colored HOMFLY through the simplest
Racah matrices
$\ \ S_R: \ \ R\otimes R \otimes \bar R \longrightarrow R \ \ $
and
$\ \ \bar S_R: \ \ R\otimes \bar R \otimes R \longrightarrow R \ \ $.

Further generalized to fingered 3-braids,
where HOMFLY are expressed through
generic Racah matrices, but more complicated mixing matrices are not involved.

Include all twisted, 2-bridge and pretzel knots.

All prime knots with up to 7 intersections are alternating, 2-bridge and thus arborescent.

8 intersections:

$\ 8_1\ - \ 8_{18}\ $ are alternating, $\ 8_{19}\ - \ 8_{21}\ $ are not.

The only non-arborescent prime knot is $8_{18}$,
it can also be beyond the 7-parametric family.

9 intersections:

$\ 9_1\ -\ 9_{41}\ $ are alternating, $\ 9_{42}\ -\ 9_{49}\ $ are not.

Non-arborescent are $\ 9_{34},\ 9_{39},\ 9_{40},\ 9_{41},\ 9_{47},\ 9_{49}$.
Of these only $9_{40}$ can be beyond the
7-parametric family.

10 intersections:

$\ 10_1\ -\ 10_{123}\ $ are alternating,
among them $\ 10_1\ -\ 10_{45}\ $ are rational
(two-bridge), $\ 10_{46}\ -\ 10_{99}\ $ also are arborescent,
$\ 10_{100}\ - \ 10_{123}\ $ are non-arborescent (polyhedral),
of these at least $\ 10_{101}\ - \ 10_{103}$, $ \ 10_{105}\ - \ 10_{108}$, $10_{110},\ 10_{111},\
10_{113},\ 10_{117}\ $
do belong to the
7-parametric family.

$\ 10_{124}\ -\ 10_{165}\ $ are non-alternating, among them
$\ 10_{124}\ -\ 10_{154}\ $ are
arborescent knots and $\ 10_{155}\ -\ 10_{165}\ $ are non-arborescent (polyhedral),
Of these only $10_{163}$ can be beyond the
7-parametric family.

11 intersections:

all the 16 mutant pairs are arborescent