L11a305

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 3 u 3 + 3 v 2 u 3 - v u 3 + 3 v 3 u 2 -8 v 2 u 2 + 5 v u 2 - u 2 - v 3 u + 5 v 2 u -8 v u + 3 u - v 2 + 3 v -2$ (db)
Jones polynomial	$-q^{19/2}+3 q^{17/2}-6 q^{15/2}+10 q^{13/2}-13 q^{11/2}+14 q^{9/2}-15 q^{7/2}+12 q^{5/2}-9 q^{3/2}+5 \sqrt{q}-\frac{3}{\sqrt{q}}+\frac{1}{q^{3/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 + z 7 a -5 - z 5 a -1 + 4 z 5 a -3 + 4 z 5 a -5 - z 5 a -7 -3 z 3 a -1 + 5 z 3 a -3 + 5 z 3 a -5 -3 z 3 a -7 - z a -1 + 3 z a -3 + 2 z a -5 -2 z a -7 + a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$- z 10 a -4 - z 10 a -6 -3 z 9 a -3 -6 z 9 a -5 -3 z 9 a -7 -4 z 8 a -2 -5 z 8 a -4 -5 z 8 a -6 -4 z 8 a -8 -3 z 7 a -1 + 5 z 7 a -3 + 15 z 7 a -5 + 3 z 7 a -7 -4 z 7 a -9 + 12 z 6 a -2 + 21 z 6 a -4 + 15 z 6 a -6 + 4 z 6 a -8 -3 z 6 a -10 - z 6 + 10 z 5 a -1 + 4 z 5 a -3 -17 z 5 a -5 -5 z 5 a -7 + 5 z 5 a -9 - z 5 a -11 -9 z 4 a -2 -25 z 4 a -4 -19 z 4 a -6 + 6 z 4 a -10 + 3 z 4 -8 z 3 a -1 -10 z 3 a -3 + 6 z 3 a -5 + 7 z 3 a -7 + z 3 a -9 + 2 z 3 a -11 + 2 z 2 a -2 + 7 z 2 a -4 + 8 z 2 a -6 + z 2 a -8 -3 z 2 a -10 - z 2 + 2 z a -1 + 5 z a -3 + z a -5 -2 z a -7 - z a -9 - z a -11 + a -4 - a -3 z -1 - a -5 z -1$ (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 =

3 is the signature of L11a305. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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