L11a338

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 3 u 3 + 4 v 2 u 3 -2 v u 3 + 3 v 3 u 2 -12 v 2 u 2 + 9 v u 2 - u 2 - v 3 u + 9 v 2 u -12 v u + 3 u -2 v 2 + 4 v -2$ (db)
Jones polynomial	$q^{3/2}-4 \sqrt{q}+\frac{9}{\sqrt{q}}-\frac{15}{q^{3/2}}+\frac{18}{q^{5/2}}-\frac{22}{q^{7/2}}+\frac{21}{q^{9/2}}-\frac{18}{q^{11/2}}+\frac{13}{q^{13/2}}-\frac{7}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z 5 a 7 + 3 z 3 a 7 + 3 z a 7 - z 7 a 5 -4 z 5 a 5 -7 z 3 a 5 -4 z a 5 + a 5 z -1 - z 7 a 3 -3 z 5 a 3 -3 z 3 a 3 -2 z a 3 - a 3 z -1 + z 5 a + 2 z 3 a + z a$ (db)
Kauffman polynomial	$- z 5 a 11 + 2 z 3 a 11 - z a 11 -3 z 6 a 10 + 5 z 4 a 10 -2 z 2 a 10 -5 z 7 a 9 + 6 z 5 a 9 - z 3 a 9 -7 z 8 a 8 + 12 z 6 a 8 -13 z 4 a 8 + 7 z 2 a 8 -6 z 9 a 7 + 7 z 7 a 7 -4 z 5 a 7 - z a 7 -2 z 10 a 6 -12 z 8 a 6 + 38 z 6 a 6 -42 z 4 a 6 + 15 z 2 a 6 -12 z 9 a 5 + 22 z 7 a 5 -9 z 5 a 5 -2 z 3 a 5 - a 5 z -1 -2 z 10 a 4 -12 z 8 a 4 + 39 z 6 a 4 -33 z 4 a 4 + 9 z 2 a 4 + a 4 -6 z 9 a 3 + 6 z 7 a 3 + 11 z 5 a 3 -11 z 3 a 3 + 4 z a 3 - a 3 z -1 -7 z 8 a 2 + 15 z 6 a 2 -7 z 4 a 2 + 2 z 2 a 2 -4 z 7 a + 9 z 5 a -6 z 3 a + 2 z a - z 6 + 2 z 4 - z 2$ (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 L11a338. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$
$r = -8$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -5$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -4$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{9}$
$r = -3$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = -2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = -1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 0$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{9}$
$r = 1$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$