L11a347

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{(t(1)-1) (t(2)-1) \left(2 t(2)^2 t(1)^2-t(2) t(1)^2+2 t(1)^2-t(2)^2 t(1)-t(1)+2 t(2)^2-t(2)+2\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db)
Jones polynomial	$-12 q^{9/2}+7 q^{7/2}-5 q^{5/2}+q^{3/2}+q^{23/2}-4 q^{21/2}+8 q^{19/2}-12 q^{17/2}+15 q^{15/2}-15 q^{13/2}+15 q^{11/2}-\sqrt{q}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$- z 7 a -5 - z 7 a -7 + z 5 a -3 -5 z 5 a -5 -3 z 5 a -7 + z 5 a -9 + 5 z 3 a -3 -11 z 3 a -5 + 2 z 3 a -9 + 8 z a -3 -14 z a -5 + 6 z a -7 + 4 a -3 z -1 -8 a -5 z -1 + 5 a -7 z -1 - a -9 z -1$ (db)
Kauffman polynomial	$z 4 a -14 + 4 z 5 a -13 -2 z 3 a -13 + 8 z 6 a -12 -8 z 4 a -12 + 2 z 2 a -12 + 10 z 7 a -11 -13 z 5 a -11 + 4 z 3 a -11 + z a -11 + 8 z 8 a -10 -9 z 6 a -10 + 3 z 2 a -10 -2 a -10 + 4 z 9 a -9 -5 z 5 a -9 -5 z 3 a -9 + 3 z a -9 + a -9 z -1 + z 10 a -8 + 6 z 8 a -8 -10 z 6 a -8 -11 z 4 a -8 + 20 z 2 a -8 -9 a -8 + 5 z 9 a -7 -10 z 7 a -7 + 3 z 5 a -7 + 3 z 3 a -7 -7 z a -7 + 5 a -7 z -1 + z 10 a -6 - z 8 a -6 + 5 z 6 a -6 -25 z 4 a -6 + 33 z 2 a -6 -14 a -6 + z 9 a -5 + z 7 a -5 -15 z 5 a -5 + 27 z 3 a -5 -21 z a -5 + 8 a -5 z -1 + z 8 a -4 -2 z 6 a -4 -5 z 4 a -4 + 14 z 2 a -4 -8 a -4 + z 7 a -3 -6 z 5 a -3 + 13 z 3 a -3 -12 z a -3 + 4 a -3 z -1$ (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 =

5 is the signature of L11a347. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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