L11a90

(Knotscape image)

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

Visit L11a90's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_18,8,19,7 X_20,10,21,9 X_22,15,5,16 X_8,20,9,19 X_16,21,17,22 X_14,12,15,11 X_10,18,11,17 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 4, -9, 8, -2, 11, -8, 5, -7, 9, -3, 6, -4, 7, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 5 v u 4 -5 u 4 -7 v u 3 + 7 u 3 + 7 v u 2 -7 u 2 -5 v u + 5 u + v -1$ (db)
Jones polynomial	$q^{17/2}-4 q^{15/2}+8 q^{13/2}-12 q^{11/2}+15 q^{9/2}-17 q^{7/2}+16 q^{5/2}-13 q^{3/2}+9 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 -2 z 5 a -1 + 4 z 5 a -3 -2 z 5 a -5 + a z 3 -7 z 3 a -1 + 7 z 3 a -3 -5 z 3 a -5 + z 3 a -7 + 3 a z -8 z a -1 + 7 z a -3 -3 z a -5 + z a -7 + 2 a z -1 -4 a -1 z -1 + 3 a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$- z 10 a -2 - z 10 a -4 -2 z 9 a -1 -6 z 9 a -3 -4 z 9 a -5 -4 z 8 a -2 -10 z 8 a -4 -8 z 8 a -6 -2 z 8 - a z 7 + 2 z 7 a -1 + 8 z 7 a -3 -5 z 7 a -5 -10 z 7 a -7 + 17 z 6 a -2 + 25 z 6 a -4 + 7 z 6 a -6 -8 z 6 a -8 + 7 z 6 + 5 a z 5 + 13 z 5 a -1 + 14 z 5 a -3 + 23 z 5 a -5 + 13 z 5 a -7 -4 z 5 a -9 -9 z 4 a -2 -9 z 4 a -4 + 3 z 4 a -6 + 8 z 4 a -8 - z 4 a -10 -6 z 4 -9 a z 3 -26 z 3 a -1 -24 z 3 a -3 -15 z 3 a -5 -6 z 3 a -7 + 2 z 3 a -9 -7 z 2 a -2 -6 z 2 a -4 -2 z 2 a -6 -2 z 2 a -8 - z 2 + 7 a z + 17 z a -1 + 13 z a -3 + 4 z a -5 + z a -7 + 3 a -2 + 3 a -4 + a -6 + 2-2 a z -1 -4 a -1 z -1 -3 a -3 z -1 - a -5 z -1$ (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 =

3 is the signature of L11a90. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 2$	$i = 4$
$r = -4$	${\mathbb Z}$
$r = -3$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 0$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{6}$
$r = 1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 4$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 6$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$