L11a335

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 5 - v 3 u 4 + 5 v 2 u 4 -4 v u 4 + 3 v 3 u 3 -12 v 2 u 3 + 11 v u 3 -3 u 3 -3 v 3 u 2 + 11 v 2 u 2 -12 v u 2 + 3 u 2 -4 v 2 u + 5 v u - u - v$ (db)
Jones polynomial	$q^{3/2}-4 \sqrt{q}+\frac{9}{\sqrt{q}}-\frac{16}{q^{3/2}}+\frac{22}{q^{5/2}}-\frac{26}{q^{7/2}}+\frac{26}{q^{9/2}}-\frac{23}{q^{11/2}}+\frac{17}{q^{13/2}}-\frac{11}{q^{15/2}}+\frac{4}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z 5 a 7 + 2 z 3 a 7 + 3 z a 7 + 2 a 7 z -1 - z 7 a 5 -3 z 5 a 5 -5 z 3 a 5 -6 z a 5 -3 a 5 z -1 - z 7 a 3 -3 z 5 a 3 -4 z 3 a 3 -2 z a 3 + a 3 z -1 + z 5 a + 2 z 3 a + z a$ (db)
Kauffman polynomial	$- z 5 a 11 + z 3 a 11 -4 z 6 a 10 + 3 z 4 a 10 -10 z 7 a 9 + 16 z 5 a 9 -13 z 3 a 9 + 6 z a 9 -13 z 8 a 8 + 20 z 6 a 8 -10 z 4 a 8 + z 2 a 8 -10 z 9 a 7 + 9 z 7 a 7 + z 5 a 7 + 3 z 3 a 7 -6 z a 7 + 2 a 7 z -1 -3 z 10 a 6 -18 z 8 a 6 + 48 z 6 a 6 -39 z 4 a 6 + 15 z 2 a 6 -3 a 6 -17 z 9 a 5 + 31 z 7 a 5 -20 z 5 a 5 + 16 z 3 a 5 -12 z a 5 + 3 a 5 z -1 -3 z 10 a 4 -12 z 8 a 4 + 39 z 6 a 4 -36 z 4 a 4 + 17 z 2 a 4 -3 a 4 -7 z 9 a 3 + 8 z 7 a 3 + 5 z 5 a 3 -7 z 3 a 3 + z a 3 + a 3 z -1 -7 z 8 a 2 + 14 z 6 a 2 -8 z 4 a 2 + 2 z 2 a 2 - a 2 -4 z 7 a + 9 z 5 a -6 z 3 a + z a - z 6 + 2 z 4 - z 2$ (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 L11a335. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$
$r = -8$	${\mathbb Z}$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{4}$
$r = -6$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -5$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -4$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -3$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = -2$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = -1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = 0$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{10}$
$r = 1$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$