L11a198

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 + 5 v 2 u 3 -8 v u 3 + 4 u 3 -8 v 2 u 2 + 15 v u 2 -8 u 2 + 4 v 2 u -8 v u + 5 u + v -1$ (db)
Jones polynomial	$q^{11/2}-4 q^{9/2}+9 q^{7/2}-15 q^{5/2}+19 q^{3/2}-23 \sqrt{q}+\frac{21}{\sqrt{q}}-\frac{19}{q^{3/2}}+\frac{14}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 3 a z 5 -4 z 5 a -1 + z 5 a -3 -3 a 3 z 3 + 8 a z 3 -8 z 3 a -1 + 2 z 3 a -3 + a 5 z -4 a 3 z + 8 a z -6 z a -1 + 2 z a -3 - a 3 z -1 + 3 a z -1 -2 a -1 z -1$ (db)
Kauffman polynomial	$-2 a 2 z 10 -2 z 10 -5 a 3 z 9 -13 a z 9 -8 z 9 a -1 -4 a 4 z 8 -10 a 2 z 8 -14 z 8 a -2 -20 z 8 - a 5 z 7 + 12 a 3 z 7 + 24 a z 7 -3 z 7 a -1 -14 z 7 a -3 + 14 a 4 z 6 + 46 a 2 z 6 + 20 z 6 a -2 -9 z 6 a -4 + 61 z 6 + 3 a 5 z 5 -2 a 3 z 5 + 10 a z 5 + 39 z 5 a -1 + 20 z 5 a -3 -4 z 5 a -5 -16 a 4 z 4 -45 a 2 z 4 -6 z 4 a -2 + 7 z 4 a -4 - z 4 a -6 -43 z 4 -3 a 5 z 3 -9 a 3 z 3 -27 a z 3 -35 z 3 a -1 -13 z 3 a -3 + z 3 a -5 + 5 a 4 z 2 + 9 a 2 z 2 -2 z 2 a -4 + 6 z 2 + a 5 z + 4 a 3 z + 11 a z + 12 z a -1 + 4 z a -3 + a 4 + 3 a 2 + 3- a 3 z -1 -3 a z -1 -2 a -1 z -1$ (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 L11a198. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = -1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{12}$
$r = 1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{9}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$