L11a190

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_14,17,15,18 X_16,5,17,6 X_4,15,5,16 X_18,13,19,14 X_22,20,7,19 X_20,12,21,11 X_12,22,13,21 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, 8, -9, 6, -3, 5, -4, 3, -6, 7, -8, 9, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 4 + u 4 -2 v 2 u 3 + 6 v u 3 -4 u 3 + 4 v 2 u 2 -9 v u 2 + 4 u 2 -4 v 2 u + 6 v u -2 u + v 2 -2 v$ (db)
Jones polynomial	$q^{5/2}-3 q^{3/2}+6 \sqrt{q}-\frac{10}{\sqrt{q}}+\frac{13}{q^{3/2}}-\frac{15}{q^{5/2}}+\frac{14}{q^{7/2}}-\frac{13}{q^{9/2}}+\frac{9}{q^{11/2}}-\frac{6}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 3 a 7 + z a 7 - z 5 a 5 - z 3 a 5 + z a 5 + 2 a 5 z -1 -2 z 5 a 3 -5 z 3 a 3 -6 z a 3 -3 a 3 z -1 - z 5 a - z 3 a + a z -1 + z 3 a -1 + z a -1$ (db)
Kauffman polynomial	$- a 6 z 10 - a 4 z 10 -3 a 7 z 9 -6 a 5 z 9 -3 a 3 z 9 -3 a 8 z 8 -5 a 6 z 8 -7 a 4 z 8 -5 a 2 z 8 - a 9 z 7 + 8 a 7 z 7 + 11 a 5 z 7 -4 a 3 z 7 -6 a z 7 + 12 a 8 z 6 + 24 a 6 z 6 + 18 a 4 z 6 + a 2 z 6 -5 z 6 + 4 a 9 z 5 - a 7 z 5 + 6 a 5 z 5 + 20 a 3 z 5 + 6 a z 5 -3 z 5 a -1 -14 a 8 z 4 -23 a 6 z 4 -4 a 4 z 4 + 11 a 2 z 4 - z 4 a -2 + 5 z 4 -5 a 9 z 3 -6 a 7 z 3 -15 a 5 z 3 -20 a 3 z 3 -3 a z 3 + 3 z 3 a -1 + 4 a 8 z 2 + 6 a 6 z 2 -6 a 4 z 2 -12 a 2 z 2 + z 2 a -2 -3 z 2 + 2 a 9 z + a 7 z + 9 a 5 z + 12 a 3 z + a z - z a -1 + 3 a 4 + 3 a 2 + 1-2 a 5 z -1 -3 a 3 z -1 - a z -1$ (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 L11a190. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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