L11a318

(Knotscape image)

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

Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_8,9,1,10 X_20,13,21,14 X_16,8,17,7 X_18,6,19,5 X_6,18,7,17 X_4,20,5,19 X_22,15,9,16 X_14,21,15,22
Gauss code	{1, -2, 3, -9, 7, -8, 6, -4}, {4, -1, 2, -3, 5, -11, 10, -6, 8, -7, 9, -5, 11, -10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 5 + v u 5 - v 3 u 4 + v 2 u 4 - v u 4 + u 4 + v 3 u 3 - v 2 u 3 + v u 3 - u 3 - v 3 u 2 + v 2 u 2 - v u 2 + u 2 + v 3 u - v 2 u + v u - u + v 2 - v$ (db)
Jones polynomial	$-q^{3/2}+\sqrt{q}-\frac{3}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{5}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{2}{q^{17/2}}+\frac{1}{q^{19/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- z 5 a 7 -4 z 3 a 7 -3 z a 7 + z 7 a 5 + 5 z 5 a 5 + 6 z 3 a 5 - a 5 z -1 + z 7 a 3 + 6 z 5 a 3 + 12 z 3 a 3 + 10 z a 3 + 3 a 3 z -1 - z 5 a -5 z 3 a -7 z a -2 a z -1$ (db)
Kauffman polynomial	$- z 2 a 12 -2 z 3 a 11 -3 z 4 a 10 + 2 z 2 a 10 -4 z 5 a 9 + 6 z 3 a 9 -5 z 6 a 8 + 12 z 4 a 8 -3 z 2 a 8 -6 z 7 a 7 + 22 z 5 a 7 -20 z 3 a 7 + 6 z a 7 -5 z 8 a 6 + 21 z 6 a 6 -23 z 4 a 6 + 8 z 2 a 6 - a 6 -3 z 9 a 5 + 12 z 7 a 5 -8 z 5 a 5 -5 z 3 a 5 + a 5 z -1 - z 10 a 4 + z 8 a 4 + 16 z 6 a 4 -37 z 4 a 4 + 21 z 2 a 4 -3 a 4 -4 z 9 a 3 + 26 z 7 a 3 -57 z 5 a 3 + 51 z 3 a 3 -19 z a 3 + 3 a 3 z -1 - z 10 a 2 + 6 z 8 a 2 -10 z 6 a 2 + z 4 a 2 + 7 z 2 a 2 -3 a 2 - z 9 a + 8 z 7 a -23 z 5 a + 28 z 3 a -13 z a + 2 a z -1$ (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 =

-5 is the signature of L11a318. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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