L11a196

(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_20,13,21,14 X_6,12,1,11 X_18,15,19,16 X_4,18,5,17 X_16,6,17,5 X_14,19,15,20 X_12,21,13,22
Gauss code	{1, -4, 2, -8, 9, -6}, {4, -1, 3, -2, 6, -11, 5, -10, 7, -9, 8, -7, 10, -5, 11, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 2 + 5 v u 2 -3 u 2 + 5 v 2 u -11 v u + 5 u -3 v 2 + 5 v -2$ (db)
Jones polynomial	$q^{11/2}-3 q^{9/2}+5 q^{7/2}-9 q^{5/2}+11 q^{3/2}-13 \sqrt{q}+\frac{12}{\sqrt{q}}-\frac{11}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z a 5 -2 z 3 a 3 -2 z a 3 + z 5 a + z 3 a + z a + z 5 a -1 + z 3 a -1 + 2 z a -1 + a -1 z -1 -2 z 3 a -3 -2 z a -3 - a -3 z -1 + z a -5$ (db)
Kauffman polynomial	$- a 2 z 10 - z 10 -3 a 3 z 9 -6 a z 9 -3 z 9 a -1 -3 a 4 z 8 -4 a 2 z 8 -4 z 8 a -2 -5 z 8 - a 5 z 7 + 10 a 3 z 7 + 18 a z 7 + 3 z 7 a -1 -4 z 7 a -3 + 13 a 4 z 6 + 26 a 2 z 6 + 3 z 6 a -2 -4 z 6 a -4 + 20 z 6 + 4 a 5 z 5 -6 a 3 z 5 -14 a z 5 - z 5 a -1 -3 z 5 a -5 -16 a 4 z 4 -32 a 2 z 4 + z 4 a -2 + 3 z 4 a -4 - z 4 a -6 -19 z 4 -4 a 5 z 3 -2 a 3 z 3 + 4 a z 3 + 5 z 3 a -1 + 7 z 3 a -3 + 4 z 3 a -5 + 6 a 4 z 2 + 11 a 2 z 2 + z 2 a -6 + 6 z 2 + a 5 z + a 3 z - a z -4 z a -1 -5 z a -3 -2 z a -5 - a -2 + a -1 z -1 + a -3 z -1$ (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 =

1 is the signature of L11a196. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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