L11n422

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_16,8,17,7 X_3,10,4,11 X_2,18,3,17 X_9,19,10,18 X_20,12,21,11 X_14,6,15,5 X_22,15,13,16 X_6,14,1,13 X_19,5,20,4 X_12,22,7,21
Gauss code	{1, -4, -3, 10, 7, -9}, {2, -1, -5, 3, 6, -11}, {9, -7, 8, -2, 4, 5, -10, -6, 11, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u v 3 - u 2 w 2 v 2 + u w 2 v 2 + 2 u v 2 + u 2 w v 2 -3 u w v 2 + w v 2 -2 v 2 + 2 u 2 w 2 v -2 u w 2 v - u v - u 2 w v + 3 u w v - w v + v + u w 2$ (db)
Jones polynomial	$q 9 -2 q 8 + 5 q 7 -6 q 6 + 8 q 5 -8 q 4 + 8 q 3 -5 q 2 + 4 q -1$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -4 - z 4 a -2 + 4 z 4 a -4 -2 z 4 a -6 - z 2 a -2 + 6 z 2 a -4 -6 z 2 a -6 + z 2 a -8 + a -2 + 3 a -4 -6 a -6 + 2 a -8 + a -4 z -2 -2 a -6 z -2 + a -8 z -2$ (db)
Kauffman polynomial	$z 9 a -5 + z 9 a -7 + 2 z 8 a -4 + 5 z 8 a -6 + 3 z 8 a -8 + z 7 a -3 + z 7 a -5 + 2 z 7 a -7 + 2 z 7 a -9 -5 z 6 a -4 -17 z 6 a -6 -11 z 6 a -8 + z 6 a -10 + z 5 a -3 -8 z 5 a -5 -15 z 5 a -7 -6 z 5 a -9 + 4 z 4 a -2 + 11 z 4 a -4 + 25 z 4 a -6 + 14 z 4 a -8 -4 z 4 a -10 + z 3 a -1 + 14 z 3 a -5 + 18 z 3 a -7 + 3 z 3 a -9 -3 z 2 a -2 -10 z 2 a -4 -23 z 2 a -6 -12 z 2 a -8 + 4 z 2 a -10 -9 z a -5 -9 z a -7 - a -2 + 5 a -4 + 11 a -6 + 5 a -8 - a -10 + 2 a -5 z -1 + 2 a -7 z -1 - a -4 z -2 -2 a -6 z -2 - a -8 z -2$ (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 L11n422. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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