L11n456

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_15,18,16,11 X_9,21,10,20 X_13,19,14,22 X_21,15,22,14 X_19,5,20,10 X_17,8,18,9 X_7,16,8,17 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {-7, 4, -6, 5}, {10, -1, -9, 8, -4, 7}, {11, -2, -5, 6, -3, 9, -8, 3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 2 + v u 2 + v 2 w u 2 - v w u 2 - v 2 x u 2 + v 2 u - v u - v 2 w u + 2 v w u + 2 v x u - v w x u + w x u - x u - w - v x + v w x - w x + x$ (db)
Jones polynomial	$-q^{5/2}-\frac{4}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 + 2 z 3 a 5 + 3 z a 5 + 2 a 5 z -1 + a 5 z -3 - z 5 a 3 -4 z 3 a 3 -9 z a 3 -7 a 3 z -1 -3 a 3 z -3 + z 5 a + 7 z 3 a + 11 z a + 8 a z -1 + 3 a z -3 - z 3 a -1 -4 z a -1 -3 a -1 z -1 - a -1 z -3$ (db)
Kauffman polynomial	$- a 5 z 9 - a 3 z 9 -3 a 6 z 8 -4 a 4 z 8 - a 2 z 8 -3 a 7 z 7 - a 5 z 7 + 3 a 3 z 7 - z 7 a -1 - a 8 z 6 + 10 a 6 z 6 + 16 a 4 z 6 + 5 a 2 z 6 + 11 a 7 z 5 + 16 a 5 z 5 + 2 a 3 z 5 + 4 a z 5 + 7 z 5 a -1 + 3 a 8 z 4 -5 a 6 z 4 -16 a 4 z 4 -4 a 2 z 4 + 4 z 4 -9 a 7 z 3 -22 a 5 z 3 -17 a 3 z 3 -18 a z 3 -14 z 3 a -1 - a 8 z 2 - a 6 z 2 -3 a 4 z 2 -15 a 2 z 2 -12 z 2 + 3 a 7 z + 12 a 5 z + 21 a 3 z + 23 a z + 11 z a -1 + 10 a 4 + 19 a 2 + 10-5 a 5 z -1 -12 a 3 z -1 -12 a z -1 -5 a -1 z -1 -3 a 4 z -2 -6 a 2 z -2 -3 z -2 + a 5 z -3 + 3 a 3 z -3 + 3 a z -3 + a -1 z -3$ (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 L11n456. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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