L11n219

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

Planar diagram presentation	X_10,1,11,2 X_8,9,1,10 X_3,12,4,13 X_22,16,9,15 X_17,3,18,2 X_4,22,5,21 X_14,5,15,6 X_13,21,14,20 X_16,12,17,11 X_19,7,20,6 X_7,19,8,18
Gauss code	{1, 5, -3, -6, 7, 10, -11, -2}, {2, -1, 9, 3, -8, -7, 4, -9, -5, 11, -10, 8, 6, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + u 3 + v -1$ (db)
Jones polynomial	$q^{13/2}-q^{11/2}+q^{9/2}-2 q^{7/2}+q^{5/2}-q^{3/2}-\frac{1}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z 5 a -3 + z 3 a -1 -5 z 3 a -3 + z 3 a -5 + 4 z a -1 -7 z a -3 + 3 z a -5 + 2 a -1 z -1 -3 a -3 z -1 + a -5 z -1$ (db)
Kauffman polynomial	$- z 9 a -3 - z 9 a -5 - z 8 a -2 -2 z 8 a -4 - z 8 a -6 + 7 z 7 a -3 + 7 z 7 a -5 + 6 z 6 a -2 + 13 z 6 a -4 + 7 z 6 a -6 - z 5 a -1 -17 z 5 a -3 -16 z 5 a -5 -10 z 4 a -2 -25 z 4 a -4 -15 z 4 a -6 + 6 z 3 a -1 + 21 z 3 a -3 + 15 z 3 a -5 + 7 z 2 a -2 + 16 z 2 a -4 + 10 z 2 a -6 + z 2 -7 z a -1 -13 z a -3 -6 z a -5 -3 a -2 -3 a -4 - a -6 + 2 a -1 z -1 + 3 a -3 z -1 + a -5 z -1$ (db)

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

-1 is the signature of L11n219. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$	$i = 2$
$r = 0$	${\mathbb Z}$	${\mathbb Z}^{2}$	${\mathbb Z}$
$r = 1$	${\mathbb Z}$
$r = 2$	${\mathbb Z}_2$	${\mathbb Z}$
$r = 3$	${\mathbb Z}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$
$r = 6$	${\mathbb Z}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$