L11a372

(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_10,11,1,12 X_4,15,5,16 X_22,17,11,18 X_18,10,19,9 X_20,8,21,7 X_8,20,9,19 X_6,22,7,21
Gauss code	{1, -2, 3, -6, 4, -11, 9, -10, 8, -5}, {5, -1, 2, -3, 6, -4, 7, -8, 10, -9, 11, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 4 + v 2 u 4 - v 4 u 3 + v 3 u 3 - v 2 u 3 + v u 3 + v 4 u 2 - v 3 u 2 + v 2 u 2 - v u 2 + u 2 + v 3 u - v 2 u + v u - u + v 2 - v$ (db)
Jones polynomial	$-q^{3/2}+\sqrt{q}-\frac{2}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{2}{q^{17/2}}+\frac{1}{q^{19/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- z 5 a 7 -4 z 3 a 7 -3 z a 7 + z 7 a 5 + 5 z 5 a 5 + 6 z 3 a 5 + z a 5 + z 7 a 3 + 6 z 5 a 3 + 11 z 3 a 3 + 7 z a 3 + a 3 z -1 - z 5 a -5 z 3 a -6 z a - a z -1$ (db)
Kauffman polynomial	$- z 2 a 12 -2 z 3 a 11 -3 z 4 a 10 + 2 z 2 a 10 -4 z 5 a 9 + 7 z 3 a 9 -2 z a 9 -4 z 6 a 8 + 9 z 4 a 8 -2 z 2 a 8 -4 z 7 a 7 + 13 z 5 a 7 -9 z 3 a 7 + 2 z a 7 -3 z 8 a 6 + 11 z 6 a 6 -8 z 4 a 6 -2 z 9 a 5 + 8 z 7 a 5 -5 z 5 a 5 -5 z 3 a 5 + 2 z a 5 - z 10 a 4 + 4 z 8 a 4 - z 6 a 4 -7 z 4 a 4 + 3 z 2 a 4 -3 z 9 a 3 + 20 z 7 a 3 -44 z 5 a 3 + 37 z 3 a 3 -11 z a 3 + a 3 z -1 - z 10 a 2 + 7 z 8 a 2 -16 z 6 a 2 + 13 z 4 a 2 -2 z 2 a 2 - a 2 - z 9 a + 8 z 7 a -22 z 5 a + 24 z 3 a -9 z a + a z -1$ (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 =

-5 is the signature of L11a372. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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