L11a222

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_12,4,13,3 X_22,10,7,9 X_20,12,21,11 X_10,22,11,21 X_16,6,17,5 X_18,16,19,15 X_14,20,15,19 X₂₇₃₈ X_4,14,5,13 X_6,18,1,17
Gauss code	{1, -9, 2, -10, 6, -11}, {9, -1, 3, -5, 4, -2, 10, -8, 7, -6, 11, -7, 8, -4, 5, -3}

A Braid Representative

A Morse Link Presentation

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

3 is the signature of L11a222. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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