L11a168

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_18,11,19,12 X_10,4,11,3 X_2,17,3,18 X_14,5,15,6 X₆₇₁₈ X_16,10,17,9 X_20,13,21,14 X_12,19,13,20 X_22,16,7,15 X_4,22,5,21
Gauss code	{1, -4, 3, -11, 5, -6}, {6, -1, 7, -3, 2, -9, 8, -5, 10, -7, 4, -2, 9, -8, 11, -10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 6 + v u 6 + 3 v 2 u 5 -4 v u 5 + u 5 -4 v 2 u 4 + 7 v u 4 -3 u 4 + 4 v 2 u 3 -7 v u 3 + 4 u 3 -3 v 2 u 2 + 7 v u 2 -4 u 2 + v 2 u -4 v u + 3 u + v -1$ (db)
Jones polynomial	$-q^{5/2}+4 q^{3/2}-8 \sqrt{q}+\frac{12}{\sqrt{q}}-\frac{18}{q^{3/2}}+\frac{19}{q^{5/2}}-\frac{20}{q^{7/2}}+\frac{18}{q^{9/2}}-\frac{13}{q^{11/2}}+\frac{8}{q^{13/2}}-\frac{4}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a 3 z 9 - a 5 z 7 + 6 a 3 z 7 - a z 7 -4 a 5 z 5 + 12 a 3 z 5 -4 a z 5 -4 a 5 z 3 + 8 a 3 z 3 -4 a z 3 - a z + a 3 z -1 - a z -1$ (db)
Kauffman polynomial	$- z 4 a 10 -4 z 5 a 9 + 2 z 3 a 9 -8 z 6 a 8 + 7 z 4 a 8 -2 z 2 a 8 -11 z 7 a 7 + 13 z 5 a 7 -4 z 3 a 7 -11 z 8 a 6 + 14 z 6 a 6 -4 z 2 a 6 -8 z 9 a 5 + 9 z 7 a 5 + 4 z 5 a 5 -2 z 3 a 5 -3 z 10 a 4 -7 z 8 a 4 + 30 z 6 a 4 -18 z 4 a 4 -14 z 9 a 3 + 42 z 7 a 3 -38 z 5 a 3 + 14 z 3 a 3 - z a 3 + a 3 z -1 -3 z 10 a 2 + 22 z 6 a 2 -23 z 4 a 2 + 4 z 2 a 2 - a 2 -6 z 9 a + 21 z 7 a -22 z 5 a + 8 z 3 a - z a + a z -1 -4 z 8 + 14 z 6 -13 z 4 + 2 z 2 - z 7 a -1 + 3 z 5 a -1 -2 z 3 a -1$ (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 =

-3 is the signature of L11a168. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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