L11n429

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_9,20,10,21 X_14,5,15,6 X_12,14,7,13 X_16,8,17,7 X_22,18,13,17 X_3,10,4,11 X_18,11,19,12 X_6,15,1,16 X_19,4,20,5 X_2,21,3,22
Gauss code	{1, -11, -7, 10, 3, -9}, {5, -1, -2, 7, 8, -4}, {4, -3, 9, -5, 6, -8, -10, 2, 11, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 w 2 u 3 + v 2 w u 3 + 2 v 2 w 2 u 2 - v w 2 u 2 - v u 2 -2 v 2 w u 2 + v w u 2 + u 2 - v 2 w 2 u + v w 2 u + v u - v w u + 2 w u -2 u - w + 1$ (db)
Jones polynomial	$- q + 3-4 q -1 + 6 q -2 -6 q -3 + 7 q -4 -5 q -5 + 5 q -6 -2 q -7 + q -8$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$z 2 a 8 + a 8 z -2 + a 8 - z 6 a 6 -5 z 4 a 6 -7 z 2 a 6 -2 a 6 z -2 -5 a 6 + z 8 a 4 + 6 z 6 a 4 + 12 z 4 a 4 + 11 z 2 a 4 + a 4 z -2 + 4 a 4 - z 6 a 2 -4 z 4 a 2 -3 z 2 a 2$ (db)
Kauffman polynomial	$z 2 a 10 + 2 z 3 a 9 + 5 z 4 a 8 -9 z 2 a 8 - a 8 z -2 + 6 a 8 + 2 z 7 a 7 -5 z 5 a 7 + 6 z 3 a 7 -6 z a 7 + 2 a 7 z -1 + 4 z 8 a 6 -18 z 6 a 6 + 31 z 4 a 6 -29 z 2 a 6 -2 a 6 z -2 + 12 a 6 + 2 z 9 a 5 -5 z 7 a 5 -3 z 5 a 5 + 10 z 3 a 5 -8 z a 5 + 2 a 5 z -1 + 7 z 8 a 4 -32 z 6 a 4 + 44 z 4 a 4 -26 z 2 a 4 - a 4 z -2 + 8 a 4 + 2 z 9 a 3 -6 z 7 a 3 -2 z 5 a 3 + 10 z 3 a 3 -3 z a 3 + 3 z 8 a 2 -14 z 6 a 2 + 18 z 4 a 2 -7 z 2 a 2 + a 2 + z 7 a -4 z 5 a + 4 z 3 a - z a$ (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 L11n429. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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