L11a122

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,3,15,4 X_18,8,19,7 X_22,20,5,19 X_20,9,21,10 X_8,21,9,22 X_16,12,17,11 X_12,16,13,15 X_10,18,11,17 X₂₅₃₆ X_4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 5, -9, 7, -8, 11, -2, 8, -7, 9, -3, 4, -5, 6, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 3 u 3 + 8 v u 2 -10 u 2 -10 v u + 8 u + 3 v -2$ (db)
Jones polynomial	$q^{9/2}-4 q^{7/2}+7 q^{5/2}-10 q^{3/2}+13 \sqrt{q}-\frac{15}{\sqrt{q}}+\frac{14}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 7 z -1 -3 z a 5 -2 a 5 z -1 + 3 z 3 a 3 + 3 z a 3 + 2 a 3 z -1 - z 5 a - z a - a z -1 - z 5 a -1 - z 3 a -1 - z a -1 + z 3 a -3$ (db)
Kauffman polynomial	$- a 2 z 10 - z 10 -2 a 3 z 9 -6 a z 9 -4 z 9 a -1 -2 a 4 z 8 -3 a 2 z 8 -6 z 8 a -2 -7 z 8 -2 a 5 z 7 + a 3 z 7 + 14 a z 7 + 7 z 7 a -1 -4 z 7 a -3 -2 a 6 z 6 -2 a 4 z 6 + 7 a 2 z 6 + 18 z 6 a -2 - z 6 a -4 + 26 z 6 - a 7 z 5 - a 5 z 5 -6 a 3 z 5 -14 a z 5 + 3 z 5 a -1 + 11 z 5 a -3 + 4 a 6 z 4 + 7 a 4 z 4 -8 a 2 z 4 -13 z 4 a -2 + 2 z 4 a -4 -26 z 4 + 3 a 7 z 3 + 9 a 5 z 3 + 13 a 3 z 3 + 6 a z 3 -6 z 3 a -1 -5 z 3 a -3 -2 a 6 z 2 -4 a 4 z 2 + 3 a 2 z 2 + 2 z 2 a -2 + 7 z 2 -3 a 7 z -8 a 5 z -9 a 3 z -3 a z + z a -1 + a 4 + a 7 z -1 + 2 a 5 z -1 + 2 a 3 z -1 + a z -1$ (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 L11a122. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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