L11a209

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_18,9,19,10 X₆₇₁₈ X_22,15,7,16 X_10,4,11,3 X_16,6,17,5 X_4,12,5,11 X_12,21,13,22 X_20,13,21,14 X_14,19,15,20 X_2,18,3,17
Gauss code	{1, -11, 5, -7, 6, -3}, {3, -1, 2, -5, 7, -8, 9, -10, 4, -6, 11, -2, 10, -9, 8, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 4 + 2 v u 4 + 5 v 2 u 3 -9 v u 3 + 2 u 3 -5 v 2 u 2 + 13 v u 2 -5 u 2 + 2 v 2 u -9 v u + 5 u + 2 v -2$ (db)
Jones polynomial	$-q^{7/2}+4 q^{5/2}-9 q^{3/2}+13 \sqrt{q}-\frac{19}{\sqrt{q}}+\frac{20}{q^{3/2}}-\frac{20}{q^{5/2}}+\frac{17}{q^{7/2}}-\frac{12}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 3 z 7 + a z 7 - a 5 z 5 + 4 a 3 z 5 + 3 a z 5 - z 5 a -1 -3 a 5 z 3 + 7 a 3 z 3 + 2 a z 3 -2 z 3 a -1 -3 a 5 z + 6 a 3 z - a z - z a -1 - a 5 z -1 + 2 a 3 z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 a 4 z 10 -2 a 2 z 10 -5 a 5 z 9 -11 a 3 z 9 -6 a z 9 -5 a 6 z 8 -6 a 4 z 8 -10 a 2 z 8 -9 z 8 -3 a 7 z 7 + 11 a 5 z 7 + 25 a 3 z 7 + 3 a z 7 -8 z 7 a -1 - a 8 z 6 + 13 a 6 z 6 + 24 a 4 z 6 + 29 a 2 z 6 -4 z 6 a -2 + 15 z 6 + 8 a 7 z 5 -12 a 5 z 5 -27 a 3 z 5 + 8 a z 5 + 14 z 5 a -1 - z 5 a -3 + 3 a 8 z 4 -12 a 6 z 4 -31 a 4 z 4 -30 a 2 z 4 + 5 z 4 a -2 -9 z 4 -5 a 7 z 3 + 11 a 5 z 3 + 18 a 3 z 3 -7 a z 3 -8 z 3 a -1 + z 3 a -3 -2 a 8 z 2 + 7 a 6 z 2 + 21 a 4 z 2 + 14 a 2 z 2 + 2 z 2 -4 a 5 z -8 a 3 z - a z + 3 z a -1 -2 a 6 -5 a 4 -3 a 2 + 1 + a 5 z -1 + 2 a 3 z -1 - a -1 z -1$ (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 =

-1 is the signature of L11a209. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -4$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -3$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 0$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{11}$
$r = 1$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{6}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$