L11n178

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_18,9,19,10 X_13,21,14,20 X_3,10,4,11 X_5,14,6,15 X_7,16,8,17 X_15,22,16,7 X_11,4,12,5 X_19,13,20,12 X_21,1,22,6 X_2,18,3,17
Gauss code	{1, -11, -4, 8, -5, 10}, {-6, -1, 2, 4, -8, 9, -3, 5, -7, 6, 11, -2, -9, 3, -10, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 + 2 v 2 u 3 -3 v u 3 + u 3 -2 v 2 u 2 + 5 v u 2 -2 u 2 + v 2 u -3 v u + 2 u + v -1$ (db)
Jones polynomial	$-q^{3/2}+3 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{7}{q^{3/2}}-\frac{9}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 3 z 7 - a 5 z 5 + 5 a 3 z 5 - a z 5 -3 a 5 z 3 + 8 a 3 z 3 -3 a z 3 -2 a 5 z + 4 a 3 z -3 a z + a 3 z -1 - a z -1$ (db)
Kauffman polynomial	$- a 5 z 9 - a 3 z 9 -3 a 6 z 8 -4 a 4 z 8 - a 2 z 8 -3 a 7 z 7 -2 a 5 z 7 + a 3 z 7 - a 8 z 6 + 8 a 6 z 6 + 8 a 4 z 6 - a 2 z 6 + 10 a 7 z 5 + 10 a 5 z 5 -6 a 3 z 5 -6 a z 5 + 3 a 8 z 4 -3 a 6 z 4 -3 a 4 z 4 -3 z 4 -8 a 7 z 3 -4 a 5 z 3 + 12 a 3 z 3 + 7 a z 3 - z 3 a -1 -2 a 8 z 2 - a 6 z 2 + a 4 z 2 + a 2 z 2 + z 2 + a 7 z - a 5 z -6 a 3 z -4 a z - a 2 + a 3 z -1 + a z -1$ (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 L11n178. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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