L11a469

(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_16,8,17,7 X_8,16,9,15 X_22,17,15,18 X_12,21,13,22 X_20,13,21,14 X_14,19,5,20 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {5, -4, 6, -3, 9, -8, 7, -6}, {10, -1, 4, -5, 11, -2, 3, -7, 8, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 4 + u 4 - v 2 u 3 + 5 v u 3 -2 v w u 3 + 2 w u 3 -3 u 3 + 3 v 2 u 2 -6 v u 2 -2 v 2 w u 2 + 6 v w u 2 -3 w u 2 + 2 u 2 -2 v 2 u + 2 v u + 3 v 2 w u -5 v w u + w u - v 2 w + v w$ (db)
Jones polynomial	$q -3 + 7 q -1 -11 q -2 + 15 q -3 -16 q -4 + 17 q -5 -13 q -6 + 11 q -7 -6 q -8 + 3 q -9 - q -10$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- a 10 + 3 z 2 a 8 + a 8 z -2 + 2 a 8 -2 z 4 a 6 - z 2 a 6 -2 a 6 z -2 -2 a 6 -3 z 4 a 4 -3 z 2 a 4 + a 4 z -2 - a 4 - z 4 a 2 + 2 z 2 a 2 + 2 a 2 + z 2$ (db)
Kauffman polynomial	$z 7 a 11 -4 z 5 a 11 + 5 z 3 a 11 -2 z a 11 + 3 z 8 a 10 -12 z 6 a 10 + 16 z 4 a 10 -9 z 2 a 10 + 3 a 10 + 3 z 9 a 9 -6 z 7 a 9 -7 z 5 a 9 + 18 z 3 a 9 -7 z a 9 + z 10 a 8 + 8 z 8 a 8 -38 z 6 a 8 + 51 z 4 a 8 -34 z 2 a 8 - a 8 z -2 + 12 a 8 + 7 z 9 a 7 -11 z 7 a 7 -13 z 5 a 7 + 25 z 3 a 7 -14 z a 7 + 2 a 7 z -1 + z 10 a 6 + 12 z 8 a 6 -38 z 6 a 6 + 39 z 4 a 6 -28 z 2 a 6 -2 a 6 z -2 + 12 a 6 + 4 z 9 a 5 + 4 z 7 a 5 -24 z 5 a 5 + 22 z 3 a 5 -10 z a 5 + 2 a 5 z -1 + 7 z 8 a 4 -6 z 6 a 4 -4 z 4 a 4 + 4 z 2 a 4 - a 4 z -2 + 2 a 4 + 8 z 7 a 3 -11 z 5 a 3 + 8 z 3 a 3 - z a 3 + 6 z 6 a 2 -7 z 4 a 2 + 6 z 2 a 2 -2 a 2 + 3 z 5 a -2 z 3 a + z 4 - z 2$ (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 L11a469. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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