L11a147

(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_18,12,19,11 X_12,6,13,5 X_4,19,5,20 X_16,7,17,8 X_20,13,21,14 X_22,15,7,16 X_14,21,15,22 X_6,18,1,17
Gauss code	{1, -2, 3, -6, 5, -11}, {7, -1, 2, -3, 4, -5, 8, -10, 9, -7, 11, -4, 6, -8, 10, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 6 + v u 6 + 2 v 2 u 5 -3 v u 5 + u 5 -2 v 2 u 4 + 3 v u 4 -2 u 4 + 2 v 2 u 3 -3 v u 3 + 2 u 3 -2 v 2 u 2 + 3 v u 2 -2 u 2 + v 2 u -3 v u + 2 u + v -1$ (db)
Jones polynomial	$\sqrt{q}-\frac{3}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{11}{q^{9/2}}+\frac{11}{q^{11/2}}-\frac{10}{q^{13/2}}+\frac{8}{q^{15/2}}-\frac{5}{q^{17/2}}+\frac{3}{q^{19/2}}-\frac{1}{q^{21/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- a 5 z 9 + a 7 z 7 -7 a 5 z 7 + a 3 z 7 + 5 a 7 z 5 -17 a 5 z 5 + 5 a 3 z 5 + 7 a 7 z 3 -16 a 5 z 3 + 6 a 3 z 3 + 2 a 7 z -2 a 5 z - a 3 z - a 7 z -1 + 3 a 5 z -1 -2 a 3 z -1$ (db)
Kauffman polynomial	$- z 3 a 13 -3 z 4 a 12 + z 2 a 12 -5 z 5 a 11 + 3 z 3 a 11 -7 z 6 a 10 + 9 z 4 a 10 -2 z 2 a 10 -8 z 7 a 9 + 16 z 5 a 9 -7 z 3 a 9 + z a 9 -7 z 8 a 8 + 16 z 6 a 8 -4 z 4 a 8 -3 z 2 a 8 + a 8 -5 z 9 a 7 + 13 z 7 a 7 -4 z 5 a 7 - z 3 a 7 - a 7 z -1 -2 z 10 a 6 + z 8 a 6 + 17 z 6 a 6 -20 z 4 a 6 + z 2 a 6 + 3 a 6 -8 z 9 a 5 + 38 z 7 a 5 -56 z 5 a 5 + 29 z 3 a 5 - z a 5 -3 a 5 z -1 -2 z 10 a 4 + 7 z 8 a 4 - z 6 a 4 -11 z 4 a 4 + 3 z 2 a 4 + 3 a 4 -3 z 9 a 3 + 17 z 7 a 3 -31 z 5 a 3 + 19 z 3 a 3 -2 a 3 z -1 - z 8 a 2 + 5 z 6 a 2 -7 z 4 a 2 + 2 z 2 a 2$ (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 =

-5 is the signature of L11a147. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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