L11a309

(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_6,9,7,10 X_20,7,21,8 X_8,19,1,20 X_22,15,9,16 X_12,4,13,3 X_16,6,17,5 X_4,14,5,13 X_14,21,15,22 X_2,18,3,17
Gauss code	{1, -11, 7, -9, 8, -3, 4, -5}, {3, -1, 2, -7, 9, -10, 6, -8, 11, -2, 5, -4, 10, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 5 + v u 5 - v 3 u 4 + 5 v 2 u 4 -5 v u 4 + u 4 + 2 v 3 u 3 -10 v 2 u 3 + 10 v u 3 -2 u 3 -2 v 3 u 2 + 10 v 2 u 2 -10 v u 2 + 2 u 2 + v 3 u -5 v 2 u + 5 v u - u + v 2 - v$ (db)
Jones polynomial	$-q^{7/2}+4 q^{5/2}-9 q^{3/2}+15 \sqrt{q}-\frac{22}{\sqrt{q}}+\frac{24}{q^{3/2}}-\frac{25}{q^{5/2}}+\frac{21}{q^{7/2}}-\frac{16}{q^{9/2}}+\frac{10}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 3 z 7 + a z 7 - a 5 z 5 + 3 a 3 z 5 + 3 a z 5 - z 5 a -1 -2 a 5 z 3 + 4 a 3 z 3 + 3 a z 3 -2 z 3 a -1 -2 a 5 z + 4 a 3 z - a z - z a -1 - a 5 z -1 + 3 a 3 z -1 -2 a z -1$ (db)
Kauffman polynomial	$-3 a 4 z 10 -3 a 2 z 10 -8 a 5 z 9 -16 a 3 z 9 -8 a z 9 -8 a 6 z 8 -11 a 4 z 8 -13 a 2 z 8 -10 z 8 -4 a 7 z 7 + 14 a 5 z 7 + 31 a 3 z 7 + 5 a z 7 -8 z 7 a -1 - a 8 z 6 + 18 a 6 z 6 + 37 a 4 z 6 + 36 a 2 z 6 -4 z 6 a -2 + 14 z 6 + 8 a 7 z 5 -7 a 5 z 5 -20 a 3 z 5 + 8 a z 5 + 12 z 5 a -1 - z 5 a -3 + 2 a 8 z 4 -13 a 6 z 4 -31 a 4 z 4 -28 a 2 z 4 + 5 z 4 a -2 -7 z 4 -4 a 7 z 3 + 7 a 5 z 3 + 10 a 3 z 3 -9 a z 3 -7 z 3 a -1 + z 3 a -3 - a 8 z 2 + 6 a 6 z 2 + 13 a 4 z 2 + 8 a 2 z 2 - z 2 a -2 + z 2 -4 a 5 z -7 a 3 z - a z + 2 z a -1 - a 6 -3 a 4 -3 a 2 + a 5 z -1 + 3 a 3 z -1 + 2 a z -1$ (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 =

-1 is the signature of L11a309. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -4$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -3$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = -2$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = -1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = 0$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{13}$
$r = 1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$