L11n39

(Knotscape image)

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

Visit L11n39's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_20,7,21,8 X_4,21,1,22 X_5,12,6,13 X₈₄₉₃ X_9,16,10,17 X_13,19,14,18 X_17,15,18,14 X_15,10,16,11 X_11,22,12,5 X_2,20,3,19
Gauss code	{1, -11, 5, -3}, {-4, -1, 2, -5, -6, 9, -10, 4, -7, 8, -9, 6, -8, 7, 11, -2, 3, 10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u + u + v -1$ (db)
Jones polynomial	$-q^{5/2}+2 q^{3/2}-2 \sqrt{q}-\frac{2}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{2}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 - a 7 z -1 + 2 z 3 a 5 + 5 z a 5 + 3 a 5 z -1 - z 5 a 3 -5 z 3 a 3 -8 z a 3 -3 a 3 z -1 + z 5 a + 5 z 3 a + 6 z a + 2 a z -1 - z 3 a -1 -2 z a -1 - a -1 z -1$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 - a 6 z 8 -2 a 4 z 8 -3 a 2 z 8 -2 z 8 -2 a 7 z 7 -3 a 5 z 7 + 4 a 3 z 7 + 4 a z 7 - z 7 a -1 - a 8 z 6 + 2 a 6 z 6 + 12 a 4 z 6 + 20 a 2 z 6 + 11 z 6 + 8 a 7 z 5 + 15 a 5 z 5 + 7 a 3 z 5 + 5 a z 5 + 5 z 5 a -1 + 4 a 8 z 4 + 5 a 6 z 4 -17 a 4 z 4 -32 a 2 z 4 -14 z 4 -7 a 7 z 3 -19 a 5 z 3 -27 a 3 z 3 -21 a z 3 -6 z 3 a -1 -4 a 8 z 2 -5 a 6 z 2 + 7 a 4 z 2 + 13 a 2 z 2 + 5 z 2 + 3 a 7 z + 12 a 5 z + 18 a 3 z + 13 a z + 4 z a -1 + a 8 + 2 a 6 -2 a 2 - a 7 z -1 -3 a 5 z -1 -3 a 3 z -1 -2 a z -1 - a -1 z -1$ (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 L11n39. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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