L11a182

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_20,9,21,10 X_6,21,1,22 X_18,8,19,7 X_10,4,11,3 X_12,16,13,15 X_14,6,15,5 X_4,14,5,13 X_16,12,17,11 X_22,18,7,17 X_2,20,3,19
Gauss code	{1, -11, 5, -8, 7, -3}, {4, -1, 2, -5, 9, -6, 8, -7, 6, -9, 10, -4, 11, -2, 3, -10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 4 + 3 v u 4 - u 4 + 5 v 2 u 3 -9 v u 3 + 4 u 3 -6 v 2 u 2 + 11 v u 2 -6 u 2 + 4 v 2 u -9 v u + 5 u - v 2 + 3 v -2$ (db)
Jones polynomial	$-q^{19/2}+4 q^{17/2}-9 q^{15/2}+15 q^{13/2}-20 q^{11/2}+23 q^{9/2}-23 q^{7/2}+19 q^{5/2}-15 q^{3/2}+8 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{1}{q^{3/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 + z 7 a -5 - z 5 a -1 + 3 z 5 a -3 + 3 z 5 a -5 - z 5 a -7 -2 z 3 a -1 + 3 z 3 a -3 + 3 z 3 a -5 -2 z 3 a -7 + z a -3 + z a -5 - z a -7 + a -1 z -1 - a -3 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -4 -2 z 10 a -6 -5 z 9 a -3 -12 z 9 a -5 -7 z 9 a -7 -6 z 8 a -2 -11 z 8 a -4 -15 z 8 a -6 -10 z 8 a -8 -4 z 7 a -1 + z 7 a -3 + 16 z 7 a -5 + 3 z 7 a -7 -8 z 7 a -9 + 11 z 6 a -2 + 28 z 6 a -4 + 36 z 6 a -6 + 16 z 6 a -8 -4 z 6 a -10 - z 6 + 10 z 5 a -1 + 16 z 5 a -3 + z 5 a -5 + 8 z 5 a -7 + 12 z 5 a -9 - z 5 a -11 -3 z 4 a -2 -15 z 4 a -4 -26 z 4 a -6 -11 z 4 a -8 + 5 z 4 a -10 + 2 z 4 -8 z 3 a -1 -13 z 3 a -3 -5 z 3 a -5 -7 z 3 a -7 -6 z 3 a -9 + z 3 a -11 - z 2 a -2 + 2 z 2 a -4 + 6 z 2 a -6 + 3 z 2 a -8 - z 2 a -10 - z 2 + z a -1 + 2 z a -3 + z a -5 + z a -7 + z a -9 - a -2 + a -1 z -1 + a -3 z -1$ (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 L11a182. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 2$	$i = 4$
$r = -3$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 0$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{6}$
$r = 1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 2$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{11}$
$r = 3$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 4$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 5$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 6$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 7$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 8$	${\mathbb Z}_2$	${\mathbb Z}$