L11n198

(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_7,17,8,16 X_15,21,16,20 X_18,14,19,13 X_21,6,22,7 X_22,18,9,17 X_19,5,20,4 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 $u$ , $v$ , $w$ , ...)	$- v 2 u 3 + v u 3 - v 3 u 2 + 2 v 2 u 2 -2 v u 2 -2 v 2 u + 2 v u - u + v 2 - v$ (db)
Jones polynomial	$q^{9/2}-2 q^{7/2}+3 q^{5/2}-4 q^{3/2}+4 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- a z 5 - z 5 a -1 + a 3 z 3 -4 a z 3 -3 z 3 a -1 + z 3 a -3 + 3 a 3 z -6 a z - z a -1 + 2 z a -3 + 2 a 3 z -1 -3 a z -1 + a -1 z -1$ (db)
Kauffman polynomial	$- a z 9 - z 9 a -1 - a 2 z 8 -2 z 8 a -2 -3 z 8 + 5 a z 7 + 3 z 7 a -1 -2 z 7 a -3 + 5 a 2 z 6 + 8 z 6 a -2 - z 6 a -4 + 14 z 6 -2 a 3 z 5 -11 a z 5 - z 5 a -1 + 8 z 5 a -3 - a 4 z 4 -11 a 2 z 4 -8 z 4 a -2 + 4 z 4 a -4 -22 z 4 - a 5 z 3 + 5 a 3 z 3 + 13 a z 3 -7 z 3 a -3 + a 4 z 2 + 8 a 2 z 2 + 4 z 2 a -2 -3 z 2 a -4 + 14 z 2 + 2 a 5 z -6 a 3 z -11 a z - z a -1 + 2 z a -3 -3 a 2 - a -2 -3 + 2 a 3 z -1 + 3 a z -1 + a -1 z -1$ (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 L11n198. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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