L11n222

(Knotscape image)

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

Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_13,20,14,21 X_5,14,6,15 X_4,21,5,22 X_16,9,17,10 X_22,15,9,16 X_17,6,18,7 X_7,18,8,19 X_19,8,20,1
Gauss code	{1, -2, 3, -6, -5, 9, -10, 11}, {7, -1, 2, -3, -4, 5, 8, -7, -9, 10, -11, 4, 6, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 5 -2 v 2 u 3 + v u 3 + v 2 u 2 -2 v u 2 -1$ (db)
Jones polynomial	$-\frac{1}{q^{9/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{17/2}}-\frac{2}{q^{19/2}}+\frac{2}{q^{21/2}}-\frac{2}{q^{23/2}}+\frac{2}{q^{25/2}}-\frac{1}{q^{27/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$- z 3 a 13 -2 z a 13 - a 13 z -1 + z 7 a 11 + 8 z 5 a 11 + 19 z 3 a 11 + 15 z a 11 + 3 a 11 z -1 - z 9 a 9 -9 z 7 a 9 -28 z 5 a 9 -36 z 3 a 9 -17 z a 9 -2 a 9 z -1$ (db)
Kauffman polynomial	$- z 3 a 17 + z a 17 -2 z 4 a 16 + 3 z 2 a 16 - z 5 a 15 + z a 15 -2 z 4 a 14 + 2 z 2 a 14 - a 14 - z 3 a 13 - z a 13 + a 13 z -1 - z 8 a 12 + 8 z 6 a 12 -19 z 4 a 12 + 14 z 2 a 12 -3 a 12 - z 9 a 11 + 9 z 7 a 11 -27 z 5 a 11 + 34 z 3 a 11 -18 z a 11 + 3 a 11 z -1 - z 8 a 10 + 8 z 6 a 10 -19 z 4 a 10 + 15 z 2 a 10 -3 a 10 - z 9 a 9 + 9 z 7 a 9 -28 z 5 a 9 + 36 z 3 a 9 -17 z a 9 + 2 a 9 z -1$ (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 =

-7 is the signature of L11n222. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -10$	$i = -8$	$i = -6$
$r = -10$		${\mathbb Z}$
$r = -9$		${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -8$		${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}$
$r = -7$		${\mathbb Z}^{2}$
$r = -6$		${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -5$	${\mathbb Z}$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$		${\mathbb Z}$	${\mathbb Z}$
$r = -3$	${\mathbb Z}$
$r = -2$	${\mathbb Z}_2$	${\mathbb Z}$
$r = -1$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$