L11a339

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{(t(1)-1) (t(2)-1) \left(t(1)^2 t(2)^4-t(1) t(2)^4+3 t(1) t(2)^3+t(1)^2 t(2)^2-t(1) t(2)^2+t(2)^2+3 t(1) t(2)-t(1)+1\right)}{t(1)^{3/2} t(2)^{5/2}}$ (db)
Jones polynomial	$14 q^{9/2}-17 q^{7/2}+16 q^{5/2}-\frac{1}{q^{5/2}}-14 q^{3/2}+\frac{3}{q^{3/2}}+q^{17/2}-3 q^{15/2}+7 q^{13/2}-11 q^{11/2}+10 \sqrt{q}-\frac{7}{\sqrt{q}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$- z 9 a -3 + z 7 a -1 -7 z 7 a -3 + z 7 a -5 + 5 z 5 a -1 -19 z 5 a -3 + 5 z 5 a -5 + 9 z 3 a -1 -25 z 3 a -3 + 9 z 3 a -5 + 8 z a -1 -15 z a -3 + 7 z a -5 + 2 a -1 z -1 -3 a -3 z -1 + a -5 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -2 -2 z 10 a -4 -4 z 9 a -1 -10 z 9 a -3 -6 z 9 a -5 - z 8 a -2 -6 z 8 a -4 -8 z 8 a -6 -3 z 8 - a z 7 + 13 z 7 a -1 + 33 z 7 a -3 + 11 z 7 a -5 -8 z 7 a -7 + 20 z 6 a -2 + 29 z 6 a -4 + 14 z 6 a -6 -6 z 6 a -8 + 11 z 6 + 4 a z 5 -11 z 5 a -1 -41 z 5 a -3 -11 z 5 a -5 + 12 z 5 a -7 -3 z 5 a -9 -24 z 4 a -2 -33 z 4 a -4 -12 z 4 a -6 + 7 z 4 a -8 - z 4 a -10 -11 z 4 -5 a z 3 + 6 z 3 a -1 + 35 z 3 a -3 + 13 z 3 a -5 -9 z 3 a -7 + 2 z 3 a -9 + 10 z 2 a -2 + 18 z 2 a -4 + 6 z 2 a -6 -4 z 2 a -8 + z 2 a -10 + 3 z 2 + 2 a z -7 z a -1 -17 z a -3 -6 z a -5 + 2 z a -7 -3 a -2 -3 a -4 - a -6 + 2 a -1 z -1 + 3 a -3 z -1 + a -5 z -1$ (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 L11a339. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 2$	$i = 4$
$r = -4$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -1$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 0$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{7}$
$r = 1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 4$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 5$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 6$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$