L11a199

(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_16,5,17,6 X_18,11,19,12 X_22,19,7,20 X_20,14,21,13 X_12,22,13,21 X_14,17,15,18 X₆₇₁₈ X_4,15,5,16
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 5, -8, 7, -9, 11, -4, 9, -5, 6, -7, 8, -6}

A Braid Representative

A Morse Link Presentation

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

-5 is the signature of L11a199. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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