L11a471

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_18,6,19,5 X₈₄₉₃ X_20,9,21,10 X_10,19,11,20 X_14,18,15,17 X_22,16,17,15 X_16,22,5,21 X_2,12,3,11
Gauss code	{1, -11, 5, -3}, {8, -4, 7, -6, 10, -9}, {4, -1, 2, -5, 6, -7, 11, -2, 3, -8, 9, -10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 4 - v w u 4 + w u 4 - u 4 + v 2 u 3 -5 v u 3 - v 2 w u 3 + 5 v w u 3 -3 w u 3 + 3 u 3 -3 v 2 u 2 + 8 v u 2 + 3 v 2 w u 2 -8 v w u 2 + 3 w u 2 -3 u 2 + 3 v 2 u -5 v u -3 v 2 w u + 5 v w u - w u + u - v 2 + v + v 2 w - v w$ (db)
Jones polynomial	$q 6 -4 q 5 + 9 q 4 -14 q 3 + 21 q 2 -22 q + 24-20 q -1 + 15 q -2 -9 q -3 + 4 q -4 - q -5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 6 a -2 - z 6 + 2 a 2 z 4 -2 z 4 a -2 + z 4 a -4 - a 4 z 2 + a 2 z 2 -4 z 2 a -2 + z 2 a -4 + 3 z 2 - a 2 -4 a -2 + a -4 + 4-2 a -2 z -2 + a -4 z -2 + z -2$ (db)
Kauffman polynomial	$2 z 10 a -2 + 2 z 10 + 6 a z 9 + 13 z 9 a -1 + 7 z 9 a -3 + 9 a 2 z 8 + 15 z 8 a -2 + 8 z 8 a -4 + 16 z 8 + 8 a 3 z 7 + 4 a z 7 -20 z 7 a -1 -12 z 7 a -3 + 4 z 7 a -5 + 4 a 4 z 6 -10 a 2 z 6 -50 z 6 a -2 -22 z 6 a -4 + z 6 a -6 -41 z 6 + a 5 z 5 -12 a 3 z 5 -21 a z 5 + 3 z 5 a -1 + 2 z 5 a -3 -9 z 5 a -5 -5 a 4 z 4 + 3 a 2 z 4 + 54 z 4 a -2 + 21 z 4 a -4 -2 z 4 a -6 + 39 z 4 - a 5 z 3 + 7 a 3 z 3 + 14 a z 3 + 6 z 3 a -1 + 4 z 3 a -3 + 4 z 3 a -5 + 2 a 4 z 2 -3 a 2 z 2 -27 z 2 a -2 -9 z 2 a -4 -23 z 2 - a 3 z -3 a z -4 z a -1 -2 z a -3 + 2 a 2 + 7 a -2 + 3 a -4 + 7 + 2 a -1 z -1 + 2 a -3 z -1 -2 a -2 z -2 - a -4 z -2 - z -2$ (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 =

0 is the signature of L11a471. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 0$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{13}$
$r = 1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{13}$
$r = 3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$