L11a323

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{(u-1) (v-1) \left(u^2 v^2-2 u^2 v+u^2-3 u v^2+5 u v-3 u+v^2-2 v+1\right)}{u^{3/2} v^{3/2}}$ (db)
Jones polynomial	$q^{9/2}-4 q^{7/2}+9 q^{5/2}-16 q^{3/2}+21 \sqrt{q}-\frac{25}{\sqrt{q}}+\frac{24}{q^{3/2}}-\frac{22}{q^{5/2}}+\frac{16}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 5 z 3 + a 5 z -2 a 3 z 5 -4 a 3 z 3 + z 3 a -3 -3 a 3 z + z a -3 + a z 7 + 3 a z 5 -2 z 5 a -1 + 5 a z 3 -4 z 3 a -1 + 4 a z -3 z a -1 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 a 2 z 10 -2 z 10 -7 a 3 z 9 -13 a z 9 -6 z 9 a -1 -10 a 4 z 8 -19 a 2 z 8 -7 z 8 a -2 -16 z 8 -8 a 5 z 7 -3 a 3 z 7 + 11 a z 7 + 2 z 7 a -1 -4 z 7 a -3 -4 a 6 z 6 + 12 a 4 z 6 + 43 a 2 z 6 + 14 z 6 a -2 - z 6 a -4 + 42 z 6 - a 7 z 5 + 11 a 5 z 5 + 23 a 3 z 5 + 20 a z 5 + 18 z 5 a -1 + 9 z 5 a -3 + 5 a 6 z 4 -4 a 4 z 4 -27 a 2 z 4 -8 z 4 a -2 + 2 z 4 a -4 -28 z 4 + a 7 z 3 -7 a 5 z 3 -20 a 3 z 3 -23 a z 3 -18 z 3 a -1 -7 z 3 a -3 -2 a 6 z 2 - a 4 z 2 + 5 a 2 z 2 + z 2 a -2 - z 2 a -4 + 6 z 2 + 2 a 5 z + 6 a 3 z + 8 a z + 6 z a -1 + 2 z a -3 + 1- a z -1 - a -1 z -1$ (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 =

-1 is the signature of L11a323. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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