L11a367

(Knotscape image)

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

Planar diagram presentation	X_12,1,13,2 X_2,13,3,14 X_14,3,15,4 X_16,5,17,6 X_18,7,19,8 X_8,11,9,12 X_20,10,21,9 X_22,20,11,19 X_10,22,1,21 X_4,15,5,16 X_6,17,7,18
Gauss code	{1, -2, 3, -10, 4, -11, 5, -6, 7, -9}, {6, -1, 2, -3, 10, -4, 11, -5, 8, -7, 9, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 4 u 4 + v 3 u 4 + v 4 u 3 -3 v 3 u 3 + v 2 u 3 + v 3 u 2 -3 v 2 u 2 + v u 2 + v 2 u -3 v u + u + v -1$ (db)
Jones polynomial	$\sqrt{q}-\frac{2}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{5}{q^{13/2}}+\frac{4}{q^{15/2}}-\frac{3}{q^{17/2}}+\frac{2}{q^{19/2}}-\frac{1}{q^{21/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- a 5 z 9 + a 7 z 7 -8 a 5 z 7 + a 3 z 7 + 6 a 7 z 5 -23 a 5 z 5 + 6 a 3 z 5 + 11 a 7 z 3 -29 a 5 z 3 + 10 a 3 z 3 + 7 a 7 z -14 a 5 z + 4 a 3 z + a 7 z -1 - a 5 z -1$ (db)
Kauffman polynomial	$- z 3 a 13 + z a 13 -2 z 4 a 12 + 2 z 2 a 12 -2 z 5 a 11 + z 3 a 11 -2 z 6 a 10 + 2 z 4 a 10 - z 2 a 10 -2 z 7 a 9 + 4 z 5 a 9 -3 z 3 a 9 + z a 9 -2 z 8 a 8 + 6 z 6 a 8 -5 z 4 a 8 + 2 z 2 a 8 -2 z 9 a 7 + 9 z 7 a 7 -15 z 5 a 7 + 16 z 3 a 7 -8 z a 7 + a 7 z -1 - z 10 a 6 + 3 z 8 a 6 + 2 z 6 a 6 -9 z 4 a 6 + 7 z 2 a 6 - a 6 -4 z 9 a 5 + 24 z 7 a 5 -48 z 5 a 5 + 42 z 3 a 5 -15 z a 5 + a 5 z -1 - z 10 a 4 + 4 z 8 a 4 -10 z 4 a 4 + 6 z 2 a 4 -2 z 9 a 3 + 13 z 7 a 3 -27 z 5 a 3 + 21 z 3 a 3 -5 z a 3 - z 8 a 2 + 6 z 6 a 2 -10 z 4 a 2 + 4 z 2 a 2$ (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 L11a367. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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