L11n82

(Knotscape image)

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

Visit L11n82's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_20,7,21,8 X_4,21,1,22 X_5,14,6,15 X_3,10,4,11 X_11,16,12,17 X_15,12,16,13 X_13,22,14,5 X_18,9,19,10 X_17,2,18,3 X_8,19,9,20
Gauss code	{1, 10, -5, -3}, {-4, -1, 2, -11, 9, 5, -6, 7, -8, 4, -7, 6, -10, -9, 11, -2, 3, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-4 v u 3 + u 3 + 6 v u 2 -4 u 2 -4 v u + 6 u + v -4$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{10}{q^{13/2}}+\frac{10}{q^{15/2}}-\frac{10}{q^{17/2}}+\frac{7}{q^{19/2}}-\frac{4}{q^{21/2}}+\frac{2}{q^{23/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- a 13 z -1 + z 3 a 11 + 3 z a 11 + 2 a 11 z -1 - z 5 a 9 -2 z 3 a 9 - z a 9 - a 9 z -1 -2 z 5 a 7 -5 z 3 a 7 - z a 7 + a 7 z -1 - z 5 a 5 -3 z 3 a 5 -3 z a 5 - a 5 z -1$ (db)
Kauffman polynomial	$-3 z 4 a 14 + 6 z 2 a 14 -3 a 14 - z 7 a 13 - z 5 a 13 + 3 z 3 a 13 -3 z a 13 + a 13 z -1 -2 z 8 a 12 + 5 z 6 a 12 -16 z 4 a 12 + 20 z 2 a 12 -7 a 12 - z 9 a 11 -7 z 5 a 11 + 14 z 3 a 11 -9 z a 11 + 2 a 11 z -1 -4 z 8 a 10 + 5 z 6 a 10 -8 z 4 a 10 + 11 z 2 a 10 -4 a 10 - z 9 a 9 -2 z 7 a 9 - z 5 a 9 + 8 z 3 a 9 -5 z a 9 + a 9 z -1 -2 z 8 a 8 -2 z 6 a 8 + 8 z 4 a 8 -3 z 2 a 8 -3 z 7 a 7 + 4 z 5 a 7 -2 z a 7 + a 7 z -1 -2 z 6 a 6 + 3 z 4 a 6 - a 6 - z 5 a 5 + 3 z 3 a 5 -3 z a 5 + a 5 z -1$ (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 L11n82. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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