L11n239

(Knotscape image)

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

Planar diagram presentation	X_12,1,13,2 X_7,16,8,17 X_5,1,6,10 X₃₇₄₆ X_9,5,10,4 X_18,14,19,13 X_22,20,11,19 X_20,15,21,16 X_14,21,15,22 X_2,11,3,12 X_17,8,18,9
Gauss code	{1, -10, -4, 5, -3, 4, -2, 11, -5, 3}, {10, -1, 6, -9, 8, 2, -11, -6, 7, -8, 9, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v u 3 - u 3 -2 v 2 u 2 + 2 v u 2 + 2 v 2 u -2 v u - v 3 + 2 v 2$ (db)
Jones polynomial	$q^{9/2}-2 q^{7/2}+2 q^{5/2}-2 q^{3/2}+\sqrt{q}-\frac{1}{\sqrt{q}}+\frac{1}{q^{5/2}}-\frac{2}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z a 5 + 2 a 5 z -1 -2 z 3 a 3 -7 z a 3 -5 a 3 z -1 + z 5 a + 6 z 3 a + 10 z a + 6 a z -1 - z 5 a -1 -5 z 3 a -1 -8 z a -1 -4 a -1 z -1 + z 3 a -3 + 2 z a -3 + a -3 z -1$ (db)
Kauffman polynomial	$- a 4 z 8 - a 2 z 8 - z 8 a -2 - z 8 - a 5 z 7 -3 a 3 z 7 -3 a z 7 -3 z 7 a -1 -2 z 7 a -3 + 5 a 4 z 6 + 6 a 2 z 6 + 3 z 6 a -2 - z 6 a -4 + 5 z 6 + 6 a 5 z 5 + 19 a 3 z 5 + 22 a z 5 + 18 z 5 a -1 + 9 z 5 a -3 -5 a 4 z 4 -6 a 2 z 4 + 4 z 4 a -2 + 4 z 4 a -4 - z 4 -11 a 5 z 3 -33 a 3 z 3 -42 a z 3 -29 z 3 a -1 -9 z 3 a -3 - a 2 z 2 -9 z 2 a -2 -3 z 2 a -4 -7 z 2 + 8 a 5 z + 22 a 3 z + 28 a z + 18 z a -1 + 4 z a -3 + a 4 + a 2 + 3 a -2 + a -4 + 3-2 a 5 z -1 -5 a 3 z -1 -6 a z -1 -4 a -1 z -1 - a -3 z -1$ (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 =

-1 is the signature of L11n239. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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