L11a219

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_12,4,13,3 X_20,10,21,9 X_22,12,7,11 X_10,22,11,21 X₂₇₃₈ X_18,15,19,16 X_6,14,1,13 X_4,18,5,17 X_16,6,17,5 X_14,19,15,20
Gauss code	{1, -6, 2, -9, 10, -8}, {6, -1, 3, -5, 4, -2, 8, -11, 7, -10, 9, -7, 11, -3, 5, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 3 v u 4 -2 u 4 + 3 v 2 u 3 -7 v u 3 + 3 u 3 -3 v 2 u 2 + 7 v u 2 -3 u 2 + 3 v 2 u -7 v u + 3 u -2 v 2 + 3 v -1$ (db)
Jones polynomial	$q^{17/2}-3 q^{15/2}+7 q^{13/2}-11 q^{11/2}+14 q^{9/2}-17 q^{7/2}+15 q^{5/2}-14 q^{3/2}+10 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 -2 z 5 a -1 + 4 z 5 a -3 -2 z 5 a -5 + a z 3 -6 z 3 a -1 + 7 z 3 a -3 -6 z 3 a -5 + z 3 a -7 + 2 a z -4 z a -1 + 7 z a -3 -6 z a -5 + 2 z a -7 + 2 a -3 z -1 -3 a -5 z -1 + a -7 z -1$ (db)
Kauffman polynomial	$- z 10 a -2 - z 10 a -4 -3 z 9 a -1 -7 z 9 a -3 -4 z 9 a -5 -7 z 8 a -2 -11 z 8 a -4 -7 z 8 a -6 -3 z 8 - a z 7 + 7 z 7 a -1 + 13 z 7 a -3 -3 z 7 a -5 -8 z 7 a -7 + 32 z 6 a -2 + 33 z 6 a -4 + 7 z 6 a -6 -6 z 6 a -8 + 12 z 6 + 4 a z 5 + 2 z 5 a -1 + 5 z 5 a -3 + 22 z 5 a -5 + 12 z 5 a -7 -3 z 5 a -9 -34 z 4 a -2 -24 z 4 a -4 + 3 z 4 a -6 + 7 z 4 a -8 - z 4 a -10 -15 z 4 -5 a z 3 -9 z 3 a -1 -12 z 3 a -3 -21 z 3 a -5 -11 z 3 a -7 + 2 z 3 a -9 + 11 z 2 a -2 + 4 z 2 a -4 -7 z 2 a -6 -5 z 2 a -8 + z 2 a -10 + 6 z 2 + 2 a z + 2 z a -1 + 5 z a -3 + 10 z a -5 + 5 z a -7 + 3 a -4 + 3 a -6 + a -8 -2 a -3 z -1 -3 a -5 z -1 - a -7 z -1$ (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 L11a219. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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