L11a472

(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_22,16,17,15 X_16,18,5,17 X_14,22,15,21 X_2,12,3,11
Gauss code	{1, -11, 5, -3}, {9, -4, 7, -6, 10, -8}, {4, -1, 2, -5, 6, -7, 11, -2, 3, -10, 8, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 4 - v u 4 - v 2 w u 4 + v w u 4 -3 v 2 u 3 + 3 v u 3 + 3 v 2 w u 3 -3 v w u 3 + w u 3 - u 3 + 3 v 2 u 2 -4 v u 2 -3 v 2 w u 2 + 4 v w u 2 -3 w u 2 + 3 u 2 - v 2 u + 3 v u + v 2 w u -3 v w u + 3 w u -3 u - v + v w - w + 1$ (db)
Jones polynomial	$q 7 -4 q 6 + 8 q 5 -12 q 4 + 17 q 3 -17 q 2 + 18 q -14 + 11 q -1 -6 q -2 + 3 q -3 - q -4$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$- z 8 a -2 -5 z 6 a -2 + z 6 a -4 + 2 z 6 - a 2 z 4 -9 z 4 a -2 + 3 z 4 a -4 + 8 z 4 -3 a 2 z 2 -8 z 2 a -2 + 2 z 2 a -4 + 9 z 2 - a 2 -4 a -2 + a -4 + 4-2 a -2 z -2 + a -4 z -2 + z -2$ (db)
Kauffman polynomial	$2 z 10 a -2 + 2 z 10 + 4 a z 9 + 11 z 9 a -1 + 7 z 9 a -3 + 3 a 2 z 8 + 10 z 8 a -2 + 10 z 8 a -4 + 3 z 8 + a 3 z 7 -13 a z 7 -34 z 7 a -1 -10 z 7 a -3 + 10 z 7 a -5 -12 a 2 z 6 -43 z 6 a -2 -15 z 6 a -4 + 8 z 6 a -6 -32 z 6 -4 a 3 z 5 + 9 a z 5 + 29 z 5 a -1 + z 5 a -3 -11 z 5 a -5 + 4 z 5 a -7 + 15 a 2 z 4 + 48 z 4 a -2 + 6 z 4 a -4 -8 z 4 a -6 + z 4 a -8 + 48 z 4 + 4 a 3 z 3 + 2 a z 3 -4 z 3 a -1 + z 3 a -3 + z 3 a -5 -2 z 3 a -7 -9 a 2 z 2 -23 z 2 a -2 -3 z 2 a -4 + 2 z 2 a -6 -27 z 2 - a 3 z -3 a z -4 z a -1 -2 z a -3 + 2 a 2 + 7 a -2 + 3 a -4 + 7 + 2 a -1 z -1 + 2 a -3 z -1 -2 a -2 z -2 - a -4 z -2 - z -2$ (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 =

2 is the signature of L11a472. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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