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