L11n439

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_11,19,12,18 X_3,11,4,10 X_9,1,10,4 X_7,15,8,14 X_13,5,14,8 X_22,19,13,20 X_20,15,21,16 X_16,21,17,22 X_17,9,18,12
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 11}, {-7, 6, 9, -10, -11, 3, 8, -9, 10, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$u v 3 - u w v 3 + x v 3 - v 3 + u w v 2 + w v 2 - u x v 2 - x v 2 - u w v - w v + u x v + x v + u w - u w x + w x - x$ (db)
Jones polynomial	$-q^{11/2}+q^{9/2}-5 q^{7/2}+2 q^{5/2}-5 q^{3/2}+\sqrt{q}-\frac{3}{\sqrt{q}}+\frac{1}{q^{3/2}}+\frac{1}{q^{5/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 5 -2 z 5 a -1 - a 3 z 3 + 7 a z 3 -11 z 3 a -1 + 3 z 3 a -3 -3 a 3 z + 15 a z -22 z a -1 + 11 z a -3 - z a -5 -3 a 3 z -1 + 14 a z -1 -22 a -1 z -1 + 14 a -3 z -1 -3 a -5 z -1 - a 3 z -3 + 5 a z -3 -9 a -1 z -3 + 7 a -3 z -3 -2 a -5 z -3$ (db)
Kauffman polynomial	$- z 8 a -2 - z 8 a -4 - a 3 z 7 - a z 7 -5 z 7 a -1 -6 z 7 a -3 - z 7 a -5 - a 4 z 6 -2 a 2 z 6 -3 z 6 a -2 + 2 z 6 a -4 -6 z 6 + 5 a 3 z 5 + 9 a z 5 + 28 z 5 a -1 + 30 z 5 a -3 + 6 z 5 a -5 + 5 a 4 z 4 + 17 a 2 z 4 + 37 z 4 a -2 + 9 z 4 a -4 + 40 z 4 -4 a 3 z 3 -18 a z 3 -46 z 3 a -1 -46 z 3 a -3 -14 z 3 a -5 -6 a 4 z 2 -34 a 2 z 2 -73 z 2 a -2 -26 z 2 a -4 -75 z 2 + 2 a 3 z + 16 a z + 37 z a -1 + 39 z a -3 + 16 z a -5 + 4 a 4 + 24 a 2 + 60 a -2 + 23 a -4 + 58-2 a 3 z -1 -12 a z -1 -24 a -1 z -1 -23 a -3 z -1 -9 a -5 z -1 - a 4 z -2 -7 a 2 z -2 -19 a -2 z -2 -7 a -4 z -2 -18 z -2 + a 3 z -3 + 5 a z -3 + 9 a -1 z -3 + 7 a -3 z -3 + 2 a -5 z -3$ (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 L11n439. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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