L11a228

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_16,6,17,5 X_20,10,21,9 X_10,22,11,21 X_18,16,19,15 X_14,20,15,19 X_2,11,3,12 X_12,3,13,4 X₄₇₅₈ X_22,14,7,13 X_6,18,1,17
Gauss code	{1, -7, 8, -9, 2, -11}, {9, -1, 3, -4, 7, -8, 10, -6, 5, -2, 11, -5, 6, -3, 4, -10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 4 + 2 v u 4 + 5 v 2 u 3 -9 v u 3 + 3 u 3 -6 v 2 u 2 + 13 v u 2 -6 u 2 + 3 v 2 u -9 v u + 5 u + 2 v -2$ (db)
Jones polynomial	$q^{15/2}-4 q^{13/2}+9 q^{11/2}-14 q^{9/2}+19 q^{7/2}-22 q^{5/2}+21 q^{3/2}-19 \sqrt{q}+\frac{13}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 - z 7 a -3 + a z 5 -4 z 5 a -1 -3 z 5 a -3 + z 5 a -5 + 3 a z 3 -8 z 3 a -1 -2 z 3 a -3 + 2 z 3 a -5 + 4 a z -8 z a -1 + 2 z a -3 + z a -5 + 2 a z -1 -3 a -1 z -1 + a -3 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -2 -2 z 10 a -4 -5 z 9 a -1 -11 z 9 a -3 -6 z 9 a -5 -10 z 8 a -2 -10 z 8 a -4 -7 z 8 a -6 -7 z 8 -6 a z 7 -3 z 7 a -1 + 15 z 7 a -3 + 8 z 7 a -5 -4 z 7 a -7 -3 a 2 z 6 + 20 z 6 a -2 + 26 z 6 a -4 + 16 z 6 a -6 - z 6 a -8 + 8 z 6 - a 3 z 5 + 10 a z 5 + 21 z 5 a -1 + 2 z 5 a -3 + z 5 a -5 + 9 z 5 a -7 + 4 a 2 z 4 -5 z 4 a -2 -11 z 4 a -4 -10 z 4 a -6 + 2 z 4 a -8 -2 z 4 + 2 a 3 z 3 -10 a z 3 -27 z 3 a -1 -9 z 3 a -3 + z 3 a -5 -5 z 3 a -7 - a 2 z 2 -8 z 2 a -2 -2 z 2 a -4 + 2 z 2 a -6 - z 2 a -8 -4 z 2 - a 3 z + 7 a z + 14 z a -1 + 5 z a -3 - z a -5 + 3 a -2 + a -4 + 3-2 a z -1 -3 a -1 z -1 - a -3 z -1$ (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 =

1 is the signature of L11a228. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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