L11n199

(Knotscape image)

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

Planar diagram presentation	X_10,1,11,2 X_12,3,13,4 X_5,14,6,15 X_16,7,17,8 X_20,15,21,16 X_13,18,14,19 X_21,6,22,7 X_17,22,18,9 X_4,19,5,20 X_2,9,3,10 X_8,11,1,12
Gauss code	{1, -10, 2, -9, -3, 7, 4, -11}, {10, -1, 11, -2, -6, 3, 5, -4, -8, 6, 9, -5, -7, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 5 - v 3 u 4 + v 2 u 4 - v u 3 - v 2 u 2 + v u - u - v$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{2}{q^{11/2}}+\frac{1}{q^{13/2}}-\frac{1}{q^{15/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$- z a 13 - a 13 z -1 + z 5 a 11 + 6 z 3 a 11 + 8 z a 11 + 3 a 11 z -1 - z 7 a 9 -6 z 5 a 9 -10 z 3 a 9 -6 z a 9 -2 a 9 z -1 - z 7 a 7 -6 z 5 a 7 -10 z 3 a 7 -5 z a 7$ (db)
Kauffman polynomial	$z 2 a 14 - a 14 + z 3 a 13 - z a 13 + a 13 z -1 - z 8 a 12 + 7 z 6 a 12 -15 z 4 a 12 + 13 z 2 a 12 -3 a 12 - z 9 a 11 + 7 z 7 a 11 -17 z 5 a 11 + 22 z 3 a 11 -14 z a 11 + 3 a 11 z -1 -2 z 8 a 10 + 12 z 6 a 10 -20 z 4 a 10 + 12 z 2 a 10 -3 a 10 - z 9 a 9 + 6 z 7 a 9 -11 z 5 a 9 + 11 z 3 a 9 -8 z a 9 + 2 a 9 z -1 - z 8 a 8 + 5 z 6 a 8 -5 z 4 a 8 - z 7 a 7 + 6 z 5 a 7 -10 z 3 a 7 + 5 z a 7$ (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 L11n199. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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