L11n449

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_13,20,14,21 X_16,12,17,11 X_19,12,20,13 X_8,16,5,15 X_14,8,15,7 X_22,17,19,18 X_18,21,9,22 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 7, -6}, {-5, 3, 9, -8}, {11, -2, 4, 5, -3, -7, 6, -4, 8, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + v w u 3 - w u 3 + u 3 + 2 v u 2 - v w u 2 + 2 w u 2 - v x u 2 - u 2 - w u + 2 v x u - v w x u + 2 w x u - x u - v x + v w x -2 w x + x$ (db)
Jones polynomial	$\sqrt{q}-\frac{5}{\sqrt{q}}+\frac{6}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{7}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{5}{q^{13/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 9 z -1 + a 9 z -3 -3 z a 7 -5 a 7 z -1 -3 a 7 z -3 + 3 z 3 a 5 + 6 z a 5 + 6 a 5 z -1 + 3 a 5 z -3 - z 5 a 3 -2 z 3 a 3 -2 z a 3 - a 3 z -1 - a 3 z -3 + z 3 a - z a - a z -1$ (db)
Kauffman polynomial	$- z 7 a 9 + 6 z 5 a 9 -13 z 3 a 9 + 13 z a 9 -6 a 9 z -1 + a 9 z -3 - z 8 a 8 + 2 z 6 a 8 + 6 z 4 a 8 -17 z 2 a 8 -3 a 8 z -2 + 13 a 8 - z 9 a 7 + 14 z 5 a 7 -29 z 3 a 7 + 27 z a 7 -14 a 7 z -1 + 3 a 7 z -3 -5 z 8 a 6 + 14 z 6 a 6 + 3 z 4 a 6 -28 z 2 a 6 -6 a 6 z -2 + 24 a 6 - z 9 a 5 -5 z 7 a 5 + 26 z 5 a 5 -30 z 3 a 5 + 22 z a 5 -12 a 5 z -1 + 3 a 5 z -3 -4 z 8 a 4 + 9 z 6 a 4 -11 z 2 a 4 -3 a 4 z -2 + 11 a 4 -6 z 7 a 3 + 18 z 5 a 3 -19 z 3 a 3 + 8 z a 3 -3 a 3 z -1 + a 3 z -3 -3 z 6 a 2 + 3 z 4 a 2 - z 2 a 2 - a 2 -5 z 3 a + a z -1 - z 2$ (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 L11n449. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$	$i = 0$
$r = -8$		${\mathbb Z}$	${\mathbb Z}$
$r = -7$		${\mathbb Z}$
$r = -6$		${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$		${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -4$		${\mathbb Z}^{8}\oplus{\mathbb Z}_2$	${\mathbb Z}^{5}$
$r = -3$		${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -2$		${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -1$		${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{4}$
$r = 1$		${\mathbb Z}_2$	${\mathbb Z}$