L11n68

(Knotscape image)

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

Visit L11n68's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_7,16,8,17 X_17,22,18,5 X_11,18,12,19 X_13,20,14,21 X_19,12,20,13 X_21,14,22,15 X_15,8,16,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, -5, 7, -6, 8, -9, 3, -4, 5, -7, 6, -8, 4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 - u 5 + u 4 + v u 3 - u 3 - v u 2 + u 2 + v u - v -1$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{15/2}}-\frac{3}{q^{17/2}}+\frac{2}{q^{19/2}}-\frac{2}{q^{21/2}}+\frac{2}{q^{23/2}}-\frac{1}{q^{25/2}}+\frac{1}{q^{27/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- z a 13 -2 a 13 z -1 + 2 z 3 a 11 + 6 z a 11 + 4 a 11 z -1 + z 3 a 9 + z a 9 - a 9 z -1 - z 7 a 7 -7 z 5 a 7 -14 z 3 a 7 -8 z a 7 - a 7 z -1$ (db)
Kauffman polynomial	$- z 6 a 16 + 5 z 4 a 16 -6 z 2 a 16 + 2 a 16 - z 7 a 15 + 4 z 5 a 15 -2 z 3 a 15 - z a 15 - z 8 a 14 + 4 z 6 a 14 -2 z 4 a 14 -2 z 2 a 14 + a 14 - z 9 a 13 + 6 z 7 a 13 -13 z 5 a 13 + 16 z 3 a 13 -10 z a 13 + 2 a 13 z -1 -2 z 8 a 12 + 13 z 6 a 12 -28 z 4 a 12 + 24 z 2 a 12 -6 a 12 - z 9 a 11 + 8 z 7 a 11 -23 z 5 a 11 + 30 z 3 a 11 -18 z a 11 + 4 a 11 z -1 - z 8 a 10 + 8 z 6 a 10 -20 z 4 a 10 + 18 z 2 a 10 -5 a 10 + z 5 a 9 -2 z 3 a 9 - z a 9 + a 9 z -1 + z 4 a 8 -2 z 2 a 8 + a 8 - z 7 a 7 + 7 z 5 a 7 -14 z 3 a 7 + 8 z a 7 - a 7 z -1$ (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 L11n68. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -8$	$i = -6$	$i = -4$
$r = -11$		${\mathbb Z}$
$r = -10$		${\mathbb Z}_2$	${\mathbb Z}$
$r = -9$		${\mathbb Z}^{2}$
$r = -8$		${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -7$	${\mathbb Z}$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}_2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -5$	${\mathbb Z}$	${\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -4$	${\mathbb Z}_2$	${\mathbb Z}^{3}$	${\mathbb Z}$
$r = -3$	${\mathbb Z}$	${\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}_2$	${\mathbb Z}$
$r = -1$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$