L11n126

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_22,10,7,9 X₂₇₃₈ X_15,5,16,4 X_5,13,6,12 X_11,16,12,17 X_6,18,1,17 X_14,20,15,19 X_20,14,21,13 X_18,21,19,22
Gauss code	{1, -4, 2, 5, -6, -8}, {4, -1, 3, -2, -7, 6, 10, -9, -5, 7, 8, -11, 9, -10, 11, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 4 - u 4 -4 v u 3 + 3 u 3 -2 v 2 u 2 + 7 v u 2 -2 u 2 + 3 v 2 u -4 v u - v 2 + v$ (db)
Jones polynomial	$q^{15/2}-3 q^{13/2}+5 q^{11/2}-8 q^{9/2}+9 q^{7/2}-10 q^{5/2}+9 q^{3/2}-7 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{2}{q^{3/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z 5 a -3 -3 z 3 a -1 + 2 z 3 a -3 -2 z 3 a -5 + 2 a z -4 z a -1 + 4 z a -3 -2 z a -5 + z a -7 + a z -1 -2 a -1 z -1 + 2 a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$- z 9 a -3 - z 9 a -5 -3 z 8 a -2 -6 z 8 a -4 -3 z 8 a -6 -3 z 7 a -1 -5 z 7 a -3 -5 z 7 a -5 -3 z 7 a -7 + 6 z 6 a -2 + 15 z 6 a -4 + 7 z 6 a -6 - z 6 a -8 - z 6 + 7 z 5 a -1 + 20 z 5 a -3 + 23 z 5 a -5 + 10 z 5 a -7 -8 z 4 a -2 -10 z 4 a -4 - z 4 a -6 + 3 z 4 a -8 -2 z 4 -3 a z 3 -15 z 3 a -1 -26 z 3 a -3 -23 z 3 a -5 -9 z 3 a -7 + 3 z 2 a -2 + 2 z 2 a -4 - z 2 a -6 -2 z 2 a -8 + 2 z 2 + 4 a z + 11 z a -1 + 13 z a -3 + 9 z a -5 + 3 z a -7 - a -2 - a z -1 -2 a -1 z -1 -2 a -3 z -1 - a -5 z -1$ (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 L11n126. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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