L11a204

(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_18,13,19,14 X_14,17,15,18 X_22,20,7,19 X_20,12,21,11 X_12,22,13,21 X₆₇₁₈ X_4,15,5,16
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 8, -9, 5, -6, 11, -4, 6, -5, 7, -8, 9, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 4 + v u 4 + 4 v 2 u 3 -5 v u 3 + u 3 -4 v 2 u 2 + 7 v u 2 -4 u 2 + v 2 u -5 v u + 4 u + v -2$ (db)
Jones polynomial	$q^{3/2}-2 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{13}{q^{7/2}}+\frac{13}{q^{9/2}}-\frac{12}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z 5 a 7 + 3 z 3 a 7 + 2 z a 7 - z 7 a 5 -4 z 5 a 5 -4 z 3 a 5 + z a 5 + 2 a 5 z -1 - z 7 a 3 -5 z 5 a 3 -10 z 3 a 3 -10 z a 3 -3 a 3 z -1 + z 5 a + 4 z 3 a + 4 z a + a z -1$ (db)
Kauffman polynomial	$- z 5 a 11 + 2 z 3 a 11 - z a 11 -3 z 6 a 10 + 6 z 4 a 10 -2 z 2 a 10 -4 z 7 a 9 + 6 z 5 a 9 -4 z 8 a 8 + 6 z 6 a 8 -3 z 4 a 8 + 2 z 2 a 8 -3 z 9 a 7 + 5 z 7 a 7 -7 z 5 a 7 + 6 z 3 a 7 -3 z a 7 - z 10 a 6 -3 z 8 a 6 + 11 z 6 a 6 -16 z 4 a 6 + 6 z 2 a 6 -5 z 9 a 5 + 14 z 7 a 5 -17 z 5 a 5 + 3 z 3 a 5 + 4 z a 5 -2 a 5 z -1 - z 10 a 4 - z 8 a 4 + 6 z 6 a 4 -5 z 4 a 4 -3 z 2 a 4 + 3 a 4 -2 z 9 a 3 + 3 z 7 a 3 + 4 z 5 a 3 -12 z 3 a 3 + 12 z a 3 -3 a 3 z -1 -2 z 8 a 2 + 3 z 6 a 2 + 6 z 4 a 2 -9 z 2 a 2 + 3 a 2 -2 z 7 a + 7 z 5 a -7 z 3 a + 4 z a - a z -1 - z 6 + 4 z 4 -4 z 2 + 1$ (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 L11a204. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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