L11a352

(Knotscape image)

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

Planar diagram presentation	X_12,1,13,2 X₈₄₉₃ X_14,5,15,6 X_20,7,21,8 X_16,13,17,14 X_18,10,19,9 X_10,11,1,12 X_6,15,7,16 X_4,21,5,22 X_22,17,11,18 X_2,20,3,19
Gauss code	{1, -11, 2, -9, 3, -8, 4, -2, 6, -7}, {7, -1, 5, -3, 8, -5, 10, -6, 11, -4, 9, -10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 3 u 3 + 5 v 2 u 3 -4 v u 3 + u 3 + 4 v 3 u 2 -13 v 2 u 2 + 11 v u 2 -3 u 2 -3 v 3 u + 11 v 2 u -13 v u + 4 u + v 3 -4 v 2 + 5 v -2$ (db)
Jones polynomial	$q^{3/2}-4 \sqrt{q}+\frac{9}{\sqrt{q}}-\frac{17}{q^{3/2}}+\frac{23}{q^{5/2}}-\frac{28}{q^{7/2}}+\frac{28}{q^{9/2}}-\frac{25}{q^{11/2}}+\frac{19}{q^{13/2}}-\frac{12}{q^{15/2}}+\frac{5}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z 5 a 7 + z 3 a 7 - z 7 a 5 -2 z 5 a 5 + 2 z a 5 + a 5 z -1 - z 7 a 3 -3 z 5 a 3 -5 z 3 a 3 -5 z a 3 - a 3 z -1 + z 5 a + 2 z 3 a + z a$ (db)
Kauffman polynomial	$- z 5 a 11 -5 z 6 a 10 + 3 z 4 a 10 -12 z 7 a 9 + 15 z 5 a 9 -5 z 3 a 9 + z a 9 -16 z 8 a 8 + 24 z 6 a 8 -11 z 4 a 8 + 3 z 2 a 8 -11 z 9 a 7 + 3 z 7 a 7 + 20 z 5 a 7 -13 z 3 a 7 + 2 z a 7 -3 z 10 a 6 -24 z 8 a 6 + 58 z 6 a 6 -36 z 4 a 6 + 6 z 2 a 6 -18 z 9 a 5 + 24 z 7 a 5 + 8 z 5 a 5 -19 z 3 a 5 + 7 z a 5 - a 5 z -1 -3 z 10 a 4 -15 z 8 a 4 + 43 z 6 a 4 -31 z 4 a 4 + 5 z 2 a 4 + a 4 -7 z 9 a 3 + 5 z 7 a 3 + 13 z 5 a 3 -18 z 3 a 3 + 8 z a 3 - a 3 z -1 -7 z 8 a 2 + 13 z 6 a 2 -7 z 4 a 2 + z 2 a 2 -4 z 7 a + 9 z 5 a -7 z 3 a + 2 z a - z 6 + 2 z 4 - z 2$ (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 =

-3 is the signature of L11a352. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$
$r = -8$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -5$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -4$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{12}$
$r = -3$	${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = -2$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{15}$	${\mathbb Z}^{15}$
$r = -1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = 0$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{11}$
$r = 1$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$