L11n78

(Knotscape image)

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

Visit L11n78's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_18,9,19,10 X_8,17,9,18 X_19,1,20,4 X_5,12,6,13 X_3,10,4,11 X_13,22,14,5 X_21,14,22,15 X_11,20,12,21 X_2,16,3,15
Gauss code	{1, -11, -7, 5}, {-6, -1, 2, -4, 3, 7, -10, 6, -8, 9, 11, -2, 4, -3, -5, 10, -9, 8}

A Braid Representative

A Morse Link Presentation

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

-3 is the signature of L11n78. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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