L11n8

(Knotscape image)

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

Visit L11n8's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_17,1,18,4 X_9,14,10,15 X₃₈₄₉ X_5,11,6,10 X_11,20,12,21 X_19,22,20,5 X_13,19,14,18 X_21,12,22,13 X_2,16,3,15
Gauss code	{1, -11, -5, 3}, {-6, -1, 2, 5, -4, 6, -7, 10, -9, 4, 11, -2, -3, 9, -8, 7, -10, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + u 3 + 3 v u 2 -3 u 2 -3 v u + 3 u + v -1$ (db)
Jones polynomial	$-q^{3/2}+2 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{6}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{2}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 - a 7 z -1 + 2 z 3 a 5 + 5 z a 5 + 3 a 5 z -1 - z 5 a 3 -4 z 3 a 3 -7 z a 3 -3 a 3 z -1 + 2 z 3 a + 4 z a + 2 a z -1 - z a -1 - a -1 z -1$ (db)
Kauffman polynomial	$- a 5 z 9 - a 3 z 9 -2 a 6 z 8 -4 a 4 z 8 -2 a 2 z 8 -2 a 7 z 7 + a 5 z 7 + 2 a 3 z 7 - a z 7 - a 8 z 6 + 7 a 6 z 6 + 18 a 4 z 6 + 10 a 2 z 6 + 8 a 7 z 5 + 10 a 5 z 5 + 6 a 3 z 5 + 4 a z 5 + 4 a 8 z 4 -3 a 6 z 4 -24 a 4 z 4 -19 a 2 z 4 -2 z 4 -7 a 7 z 3 -19 a 5 z 3 -21 a 3 z 3 -10 a z 3 - z 3 a -1 -3 a 8 z 2 -3 a 6 z 2 + 8 a 4 z 2 + 11 a 2 z 2 + 3 z 2 + 3 a 7 z + 12 a 5 z + 16 a 3 z + 9 a z + 2 z a -1 + a 8 + 2 a 6 -2 a 2 - a 7 z -1 -3 a 5 z -1 -3 a 3 z -1 -2 a z -1 - a -1 z -1$ (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 L11n8. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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