L11a390

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_16,7,17,8 X_8,15,5,16 X_18,12,19,11 X_20,13,21,14 X_22,20,9,19 X_12,21,13,22 X_14,18,15,17 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, 5, -8, 6, -9, 4, -3, 9, -5, 7, -6, 8, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v u 3 - v w u 3 + 2 w u 3 -2 u 3 -7 v u 2 + 4 v w u 2 -7 w u 2 + 6 u 2 + 7 v u -6 v w u + 7 w u -4 u -2 v + 2 v w -2 w + 1$ (db)
Jones polynomial	$- q 2 + 5 q -10 + 15 q -1 -18 q -2 + 21 q -3 -19 q -4 + 16 q -5 -10 q -6 + 6 q -7 -2 q -8 + q -9$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a 10 z -2 -3 a 8 z -2 -4 a 8 + 6 z 2 a 6 + 4 a 6 z -2 + 8 a 6 -4 z 4 a 4 -7 z 2 a 4 -3 a 4 z -2 -7 a 4 + z 6 a 2 + 2 z 4 a 2 + 4 z 2 a 2 + a 2 z -2 + 3 a 2 - z 4$ (db)
Kauffman polynomial	$z 6 a 10 -4 z 4 a 10 + 6 z 2 a 10 + a 10 z -2 -4 a 10 + 2 z 7 a 9 -5 z 5 a 9 + 3 z 3 a 9 + z a 9 - a 9 z -1 + 2 z 8 a 8 + 2 z 6 a 8 -18 z 4 a 8 + 25 z 2 a 8 + 3 a 8 z -2 -14 a 8 + 2 z 9 a 7 + 3 z 7 a 7 -11 z 5 a 7 + 6 z 3 a 7 + z a 7 - a 7 z -1 + z 10 a 6 + 6 z 8 a 6 -5 z 6 a 6 -19 z 4 a 6 + 35 z 2 a 6 + 4 a 6 z -2 -21 a 6 + 7 z 9 a 5 -3 z 7 a 5 -12 z 5 a 5 + 10 z 3 a 5 + z a 5 - a 5 z -1 + z 10 a 4 + 14 z 8 a 4 -23 z 6 a 4 -3 z 4 a 4 + 23 z 2 a 4 + 3 a 4 z -2 -14 a 4 + 5 z 9 a 3 + 6 z 7 a 3 -22 z 5 a 3 + 10 z 3 a 3 + z a 3 - a 3 z -1 + 10 z 8 a 2 -12 z 6 a 2 -3 z 4 a 2 + 7 z 2 a 2 + a 2 z -2 -4 a 2 + 10 z 7 a -15 z 5 a + 3 z 3 a + 5 z 6 -5 z 4 + z 5 a -1$ (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 =

-2 is the signature of L11a390. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -3$	$i = -1$
$r = -8$	${\mathbb Z}$
$r = -7$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -4$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{8}$
$r = -3$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = -1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 0$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{9}$
$r = 1$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$