L11a374

(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_22,7,11,8 X_18,10,19,9 X_20,18,21,17 X_10,20,1,19 X_8,11,9,12 X_4,15,5,16 X_6,21,7,22
Gauss code	{1, -2, 3, -10, 4, -11, 5, -9, 6, -8}, {9, -1, 2, -3, 10, -4, 7, -6, 8, -7, 11, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 4 u 4 + v 3 u 4 + v 4 u 3 -4 v 3 u 3 + 2 v 2 u 3 + 2 v 3 u 2 -5 v 2 u 2 + 2 v u 2 + 2 v 2 u -4 v u + u + v -1$ (db)
Jones polynomial	$\sqrt{q}-\frac{2}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{6}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{7}{q^{13/2}}+\frac{6}{q^{15/2}}-\frac{4}{q^{17/2}}+\frac{2}{q^{19/2}}-\frac{1}{q^{21/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- a 5 z 9 + a 7 z 7 -8 a 5 z 7 + a 3 z 7 + 6 a 7 z 5 -24 a 5 z 5 + 6 a 3 z 5 + 12 a 7 z 3 -33 a 5 z 3 + 11 a 3 z 3 + 9 a 7 z -18 a 5 z + 6 a 3 z + a 7 z -1 - a 5 z -1$ (db)
Kauffman polynomial	$- z 3 a 13 + z a 13 -2 z 4 a 12 + z 2 a 12 -3 z 5 a 11 + 2 z 3 a 11 - z a 11 -4 z 6 a 10 + 6 z 4 a 10 -4 z 2 a 10 -4 z 7 a 9 + 7 z 5 a 9 -2 z 3 a 9 -4 z 8 a 8 + 11 z 6 a 8 -9 z 4 a 8 + 5 z 2 a 8 -3 z 9 a 7 + 10 z 7 a 7 -12 z 5 a 7 + 15 z 3 a 7 -9 z a 7 + a 7 z -1 - z 10 a 6 - z 8 a 6 + 18 z 6 a 6 -29 z 4 a 6 + 16 z 2 a 6 - a 6 -5 z 9 a 5 + 26 z 7 a 5 -46 z 5 a 5 + 40 z 3 a 5 -18 z a 5 + a 5 z -1 - z 10 a 4 + 2 z 8 a 4 + 9 z 6 a 4 -23 z 4 a 4 + 12 z 2 a 4 -2 z 9 a 3 + 12 z 7 a 3 -24 z 5 a 3 + 20 z 3 a 3 -7 z a 3 - z 8 a 2 + 6 z 6 a 2 -11 z 4 a 2 + 6 z 2 a 2$ (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 L11a374. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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