L11a189

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 4 + 2 u 4 -2 v 2 u 3 + 7 v u 3 -3 u 3 + 3 v 2 u 2 -9 v u 2 + 3 u 2 -3 v 2 u + 7 v u -2 u + 2 v 2 -2 v$ (db)
Jones polynomial	$-\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{14}{q^{7/2}}+\frac{15}{q^{9/2}}-\frac{15}{q^{11/2}}+\frac{12}{q^{13/2}}-\frac{9}{q^{15/2}}+\frac{5}{q^{17/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z 3 a 9 -2 z a 9 - a 9 z -1 + z 5 a 7 + 2 z 3 a 7 + 3 z a 7 + 2 a 7 z -1 + 2 z 5 a 5 + 4 z 3 a 5 + 2 z a 5 + z 5 a 3 -3 z a 3 - a 3 z -1 - z 3 a - z a$ (db)
Kauffman polynomial	$- z 6 a 12 + 4 z 4 a 12 -4 z 2 a 12 -2 z 7 a 11 + 6 z 5 a 11 -4 z 3 a 11 -3 z 8 a 10 + 9 z 6 a 10 -11 z 4 a 10 + 9 z 2 a 10 -2 a 10 -3 z 9 a 9 + 9 z 7 a 9 -17 z 5 a 9 + 20 z 3 a 9 -7 z a 9 + a 9 z -1 - z 10 a 8 -4 z 8 a 8 + 17 z 6 a 8 -28 z 4 a 8 + 22 z 2 a 8 -5 a 8 -6 z 9 a 7 + 15 z 7 a 7 -22 z 5 a 7 + 19 z 3 a 7 -10 z a 7 + 2 a 7 z -1 - z 10 a 6 -6 z 8 a 6 + 17 z 6 a 6 -20 z 4 a 6 + 9 z 2 a 6 -3 a 6 -3 z 9 a 5 - z 7 a 5 + 11 z 5 a 5 -13 z 3 a 5 + 3 z a 5 -5 z 8 a 4 + 7 z 6 a 4 -2 z 4 a 4 - z 2 a 4 + a 4 -5 z 7 a 3 + 9 z 5 a 3 -6 z 3 a 3 + 5 z a 3 - a 3 z -1 -3 z 6 a 2 + 5 z 4 a 2 - z 2 a 2 - z 5 a + 2 z 3 a - z a$ (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 L11a189. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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