L11n441

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_18,11,19,12 X_10,3,11,4 X_4,9,1,10 X_14,7,15,8 X_8,13,5,14 X_19,13,20,22 X_15,21,16,20 X_21,17,22,16 X_12,17,9,18
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 $u$ , $v$ , $w$ , ...)	$v u 3 - v w u 3 + w u 3 - v x u 3 - w x u 3 + 2 x u 3 - u 3 - v u 2 + 2 v w u 2 - w u 2 + v x u 2 + w x u 2 -2 x u 2 + v u -2 v w u + w u - v x u - w x u + 2 x u - v + 2 v w - w + v x - v w x + w x - x$ (db)
Jones polynomial	$3 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{9}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{12}{q^{9/2}}+\frac{6}{q^{11/2}}-\frac{6}{q^{13/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 9 z -1 + 2 a 9 z -3 -4 z a 7 -11 a 7 z -1 -7 a 7 z -3 + 6 z 3 a 5 + 20 z a 5 + 23 a 5 z -1 + 9 a 5 z -3 -3 z 5 a 3 -13 z 3 a 3 -22 z a 3 -17 a 3 z -1 -5 a 3 z -3 + 3 z 3 a + 6 z a + 4 a z -1 + a z -3$ (db)
Kauffman polynomial	$- z 7 a 9 + 6 z 5 a 9 -14 z 3 a 9 + 16 z a 9 -9 a 9 z -1 + 2 a 9 z -3 - z 8 a 8 + z 6 a 8 + 10 z 4 a 8 -26 z 2 a 8 -7 a 8 z -2 + 23 a 8 - z 9 a 7 -2 z 7 a 7 + 21 z 5 a 7 -41 z 3 a 7 + 39 z a 7 -23 a 7 z -1 + 7 a 7 z -3 -6 z 8 a 6 + 10 z 6 a 6 + 29 z 4 a 6 -73 z 2 a 6 -19 a 6 z -2 + 60 a 6 - z 9 a 5 -12 z 7 a 5 + 47 z 5 a 5 -51 z 3 a 5 + 37 z a 5 -24 a 5 z -1 + 9 a 5 z -3 -5 z 8 a 4 - z 6 a 4 + 44 z 4 a 4 -75 z 2 a 4 -18 a 4 z -2 + 58 a 4 -11 z 7 a 3 + 29 z 5 a 3 -27 z 3 a 3 + 16 z a 3 -12 a 3 z -1 + 5 a 3 z -3 -10 z 6 a 2 + 25 z 4 a 2 -34 z 2 a 2 -7 a 2 z -2 + 24 a 2 -3 z 5 a -3 z 3 a + 2 z a -2 a z -1 + a z -3 -6 z 2 - z -2 + 4$ (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 L11n441. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$	$i = 0$
$r = -8$		${\mathbb Z}$	${\mathbb Z}$
$r = -7$		${\mathbb Z}$
$r = -6$		${\mathbb Z}^{5}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$		${\mathbb Z}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -4$		${\mathbb Z}^{11}\oplus{\mathbb Z}_2$	${\mathbb Z}^{7}$
$r = -3$		${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -2$		${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -1$		${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 1$		${\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$