L11a211

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_16,5,17,6 X_14,7,15,8 X_22,15,7,16 X_20,18,21,17 X_18,12,19,11 X_12,20,13,19 X_4,13,5,14 X_6,21,1,22
Gauss code	{1, -2, 3, -10, 4, -11}, {5, -1, 2, -3, 8, -9, 10, -5, 6, -4, 7, -8, 9, -7, 11, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 4 + 2 v u 4 + 5 v 2 u 3 -8 v u 3 + 2 u 3 -6 v 2 u 2 + 11 v u 2 -6 u 2 + 2 v 2 u -8 v u + 5 u + 2 v -2$ (db)
Jones polynomial	$q^{3/2}-4 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{13}{q^{3/2}}+\frac{17}{q^{5/2}}-\frac{20}{q^{7/2}}+\frac{19}{q^{9/2}}-\frac{17}{q^{11/2}}+\frac{12}{q^{13/2}}-\frac{7}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z 5 a 7 + 3 z 3 a 7 + 3 z a 7 + a 7 z -1 - z 7 a 5 -4 z 5 a 5 -7 z 3 a 5 -6 z a 5 - a 5 z -1 - z 7 a 3 -3 z 5 a 3 -2 z 3 a 3 + z 5 a + 2 z 3 a$ (db)
Kauffman polynomial	$- z 5 a 11 + 2 z 3 a 11 - z a 11 -3 z 6 a 10 + 5 z 4 a 10 -2 z 2 a 10 -5 z 7 a 9 + 7 z 5 a 9 -3 z 3 a 9 + z a 9 -6 z 8 a 8 + 8 z 6 a 8 -5 z 4 a 8 + 2 z 2 a 8 -5 z 9 a 7 + 5 z 7 a 7 -4 z 5 a 7 + 5 z 3 a 7 -4 z a 7 + a 7 z -1 -2 z 10 a 6 -7 z 8 a 6 + 21 z 6 a 6 -22 z 4 a 6 + 9 z 2 a 6 - a 6 -11 z 9 a 5 + 26 z 7 a 5 -27 z 5 a 5 + 16 z 3 a 5 -6 z a 5 + a 5 z -1 -2 z 10 a 4 -8 z 8 a 4 + 30 z 6 a 4 -28 z 4 a 4 + 8 z 2 a 4 -6 z 9 a 3 + 12 z 7 a 3 -5 z 5 a 3 + z 3 a 3 -7 z 8 a 2 + 19 z 6 a 2 -14 z 4 a 2 + 3 z 2 a 2 -4 z 7 a + 10 z 5 a -5 z 3 a - z 6 + 2 z 4$ (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 =

-3 is the signature of L11a211. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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