L11n244

(Knotscape image)

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

Planar diagram presentation	X_12,1,13,2 X_10,11,1,12 X_5,15,6,14 X_9,19,10,18 X_17,3,18,2 X_7,16,8,17 X₃₈₄₉ X_15,20,16,21 X_22,13,11,14 X_19,4,20,5 X_21,7,22,6
Gauss code	{1, 5, -7, 10, -3, 11, -6, 7, -4, -2}, {2, -1, 9, 3, -8, 6, -5, 4, -10, 8, -11, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	0 (db)
Jones polynomial	$-q^{5/2}+2 q^{3/2}-2 \sqrt{q}-\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z 3 a 5 + 2 z a 5 - z 5 a 3 -5 z 3 a 3 -6 z a 3 + z 5 a + 5 z 3 a + 6 z a + a z -1 - z 3 a -1 -2 z a -1 - a -1 z -1$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 - a 4 z 8 -3 a 2 z 8 -2 z 8 - a 5 z 7 + 4 a 3 z 7 + 4 a z 7 - z 7 a -1 -2 a 6 z 6 + 5 a 4 z 6 + 18 a 2 z 6 + 11 z 6 - a 7 z 5 + 5 a 5 z 5 + 5 a 3 z 5 + 4 a z 5 + 5 z 5 a -1 + 8 a 6 z 4 -2 a 4 z 4 -24 a 2 z 4 -14 z 4 + 3 a 7 z 3 -5 a 5 z 3 -21 a 3 z 3 -19 a z 3 -6 z 3 a -1 -4 a 6 z 2 -4 a 4 z 2 + 4 a 2 z 2 + 4 z 2 + 4 a 5 z + 12 a 3 z + 12 a z + 4 z a -1 + 1- a z -1 - a -1 z -1$ (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 =

-2 is the signature of L11n244. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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