L11a439

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,6,15,5 X_10,4,11,3 X_2,16,3,15 X_20,18,21,17 X_16,9,17,10 X_8,20,9,19 X_18,8,19,7 X_22,12,13,11 X_12,14,5,13 X_4,21,1,22
Gauss code	{1, -4, 3, -11}, {2, -1, 8, -7, 6, -3, 9, -10}, {10, -2, 4, -6, 5, -8, 7, -5, 11, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 3 u 3 -2 v 2 u 3 + v u 3 - v 3 w u 3 + 2 v 2 w u 3 - v w u 3 -2 v 3 u 2 + 5 v 2 u 2 -4 v u 2 + 2 v 3 w u 2 -5 v 2 w u 2 + 4 v w u 2 - w u 2 + u 2 + v 3 u -4 v 2 u + 5 v u - v 3 w u + 4 v 2 w u -5 v w u + 2 w u -2 u + v 2 -2 v - v 2 w + 2 v w - w + 1$ (db)
Jones polynomial	$q 10 -4 q 9 + 9 q 8 -14 q 7 + 18 q 6 -20 q 5 + 21 q 4 -16 q 3 + 13 q 2 -7 q + 4- q -1$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z 8 a -4 - z 6 a -2 + 5 z 6 a -4 -2 z 6 a -6 -3 z 4 a -2 + 9 z 4 a -4 -7 z 4 a -6 + z 4 a -8 - z 2 a -2 + 6 z 2 a -4 -7 z 2 a -6 + 2 z 2 a -8 + 2 a -2 - a -4 -2 a -6 + a -8 + a -2 z -2 -2 a -4 z -2 + a -6 z -2$ (db)
Kauffman polynomial	$2 z 10 a -4 + 2 z 10 a -6 + 5 z 9 a -3 + 12 z 9 a -5 + 7 z 9 a -7 + 4 z 8 a -2 + 7 z 8 a -4 + 15 z 8 a -6 + 12 z 8 a -8 + z 7 a -1 -15 z 7 a -3 -28 z 7 a -5 + z 7 a -7 + 13 z 7 a -9 -15 z 6 a -2 -40 z 6 a -4 -49 z 6 a -6 -15 z 6 a -8 + 9 z 6 a -10 -3 z 5 a -1 + 10 z 5 a -3 + 8 z 5 a -5 -27 z 5 a -7 -18 z 5 a -9 + 4 z 5 a -11 + 17 z 4 a -2 + 46 z 4 a -4 + 38 z 4 a -6 -8 z 4 a -10 + z 4 a -12 + 2 z 3 a -1 + z 3 a -3 + 10 z 3 a -5 + 21 z 3 a -7 + 9 z 3 a -9 - z 3 a -11 -5 z 2 a -2 -15 z 2 a -4 -9 z 2 a -6 + 4 z 2 a -8 + 3 z 2 a -10 + 2 z a -3 -2 z a -5 -6 z a -7 -2 z a -9 -2 a -2 -2 a -4 - a -6 - a -8 - a -10 -2 a -3 z -1 -2 a -5 z -1 + a -2 z -2 + 2 a -4 z -2 + a -6 z -2$ (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 =

4 is the signature of L11a439. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 3$	$i = 5$
$r = -3$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 0$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{6}$
$r = 1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 2$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 3$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 4$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{10}$
$r = 5$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 6$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 7$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 8$	${\mathbb Z}_2$	${\mathbb Z}$