L11a470

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 + 3 v 2 u 3 -4 v u 3 -2 v 2 w u 3 + 3 v w u 3 - w u 3 + 2 u 3 -3 v 2 u 2 + 6 v u 2 + 4 v 2 w u 2 -6 v w u 2 + 3 w u 2 -4 u 2 + v 2 u -3 v u -2 v 2 w u + 4 v w u -3 w u + 2 u - v w + w$ (db)
Jones polynomial	$1-3 q -1 + 8 q -2 -12 q -3 + 17 q -4 -19 q -5 + 20 q -6 -16 q -7 + 13 q -8 -7 q -9 + 3 q -10 - q -11$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- z 2 a 10 -2 a 10 + 3 z 4 a 8 + 8 z 2 a 8 + a 8 z -2 + 6 a 8 -2 z 6 a 6 -7 z 4 a 6 -10 z 2 a 6 -2 a 6 z -2 -8 a 6 - z 6 a 4 - z 4 a 4 + 3 z 2 a 4 + a 4 z -2 + 3 a 4 + z 4 a 2 + 2 z 2 a 2 + a 2$ (db)
Kauffman polynomial	$z 5 a 13 -2 z 3 a 13 + z a 13 + 3 z 6 a 12 -5 z 4 a 12 + 3 z 2 a 12 - a 12 + 5 z 7 a 11 -6 z 5 a 11 + z 3 a 11 + z a 11 + 6 z 8 a 10 -7 z 6 a 10 + 4 z 4 a 10 -2 z 2 a 10 + 4 z 9 a 9 + 3 z 7 a 9 -15 z 5 a 9 + 14 z 3 a 9 -3 z a 9 + z 10 a 8 + 14 z 8 a 8 -38 z 6 a 8 + 46 z 4 a 8 -31 z 2 a 8 - a 8 z -2 + 10 a 8 + 8 z 9 a 7 -7 z 7 a 7 -12 z 5 a 7 + 19 z 3 a 7 -10 z a 7 + 2 a 7 z -1 + z 10 a 6 + 13 z 8 a 6 -40 z 6 a 6 + 48 z 4 a 6 -36 z 2 a 6 -2 a 6 z -2 + 14 a 6 + 4 z 9 a 5 -2 z 7 a 5 -11 z 5 a 5 + 12 z 3 a 5 -7 z a 5 + 2 a 5 z -1 + 5 z 8 a 4 -11 z 6 a 4 + 8 z 4 a 4 -7 z 2 a 4 - a 4 z -2 + 5 a 4 + 3 z 7 a 3 -7 z 5 a 3 + 4 z 3 a 3 + z 6 a 2 -3 z 4 a 2 + 3 z 2 a 2 - a 2$ (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 =

-4 is the signature of L11a470. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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