L11n169

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_14,5,15,6 X_15,20,16,21 X_11,18,12,19 X_19,12,20,13 X_17,22,18,7 X_21,16,22,17 X₆₇₁₈ X_4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, -6, 7, 11, -4, -5, 9, -8, 6, -7, 5, -9, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 6 + v 2 u 4 -2 v u 4 - v 2 u 3 + 3 v u 3 - u 3 -2 v u 2 + u 2 -1$ (db)
Jones polynomial	$-\frac{1}{q^{9/2}}-\frac{1}{q^{13/2}}-\frac{1}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{4}{q^{21/2}}-\frac{4}{q^{23/2}}+\frac{3}{q^{25/2}}-\frac{2}{q^{27/2}}+\frac{1}{q^{29/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-2 z 3 a 13 -5 z a 13 -2 a 13 z -1 + z 7 a 11 + 9 z 5 a 11 + 24 z 3 a 11 + 22 z a 11 + 5 a 11 z -1 - z 9 a 9 -9 z 7 a 9 -28 z 5 a 9 -37 z 3 a 9 -20 z a 9 -3 a 9 z -1$ (db)
Kauffman polynomial	$- z 4 a 18 + 2 z 2 a 18 -2 z 5 a 17 + 4 z 3 a 17 - z a 17 -2 z 6 a 16 + 4 z 4 a 16 -3 z 2 a 16 + a 16 - z 7 a 15 + z 5 a 15 - z 3 a 15 -2 z 6 a 14 + 3 z 4 a 14 -3 z 2 a 14 -2 z 5 a 13 + 3 z 3 a 13 -5 z a 13 + 2 a 13 z -1 - z 8 a 12 + 9 z 6 a 12 -26 z 4 a 12 + 24 z 2 a 12 -5 a 12 - z 9 a 11 + 10 z 7 a 11 -33 z 5 a 11 + 45 z 3 a 11 -26 z a 11 + 5 a 11 z -1 - z 8 a 10 + 9 z 6 a 10 -24 z 4 a 10 + 22 z 2 a 10 -5 a 10 - z 9 a 9 + 9 z 7 a 9 -28 z 5 a 9 + 37 z 3 a 9 -20 z a 9 + 3 a 9 z -1$ (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 =

-7 is the signature of L11n169. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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