L11n81

(Knotscape image)

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

Visit L11n81's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_20,9,21,10 X_8,19,9,20 X_4,21,1,22 X_5,14,6,15 X_10,4,11,3 X_11,16,12,17 X_15,12,16,13 X_13,22,14,5 X_2,18,3,17
Gauss code	{1, -11, 7, -5}, {-6, -1, 2, -4, 3, -7, -8, 9, -10, 6, -9, 8, 11, -2, 4, -3, 5, 10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 2 v u 4 -2 u 4 -3 v u 3 + 3 u 3 + 3 v u 2 -3 u 2 -2 v u + 2 u + v -1$ (db)
Jones polynomial	$-\frac{1}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{5}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{7}{q^{13/2}}+\frac{6}{q^{15/2}}-\frac{4}{q^{17/2}}+\frac{2}{q^{19/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- z a 9 - a 9 z -1 - z 5 a 7 -2 z 3 a 7 + z a 7 + a 7 z -1 + z 7 a 5 + 5 z 5 a 5 + 8 z 3 a 5 + 5 z a 5 + 2 a 5 z -1 - z 5 a 3 -4 z 3 a 3 -5 z a 3 -2 a 3 z -1$ (db)
Kauffman polynomial	$-3 z 2 a 12 + a 12 - z 5 a 11 -3 z 3 a 11 + z a 11 -3 z 6 a 10 + 3 z 4 a 10 -2 z 2 a 10 -4 z 7 a 9 + 8 z 5 a 9 -6 z 3 a 9 + 2 z a 9 + a 9 z -1 -3 z 8 a 8 + 5 z 6 a 8 - z 4 a 8 + 6 z 2 a 8 -3 a 8 - z 9 a 7 -4 z 7 a 7 + 21 z 5 a 7 -19 z 3 a 7 + 4 z a 7 + a 7 z -1 -5 z 8 a 6 + 16 z 6 a 6 -11 z 4 a 6 + 2 z 2 a 6 - z 9 a 5 - z 7 a 5 + 17 z 5 a 5 -25 z 3 a 5 + 10 z a 5 -2 a 5 z -1 -2 z 8 a 4 + 8 z 6 a 4 -7 z 4 a 4 -3 z 2 a 4 + 3 a 4 - z 7 a 3 + 5 z 5 a 3 -9 z 3 a 3 + 7 z a 3 -2 a 3 z -1$ (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 =

-5 is the signature of L11n81. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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