L11n427

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_11,18,12,19 X_3,10,4,11 X_19,2,20,3 X_7,16,8,17 X_20,9,21,10 X_17,12,18,7 X_22,16,13,15 X_14,6,15,5 X_4,14,5,13 X_6,21,1,22
Gauss code	{1, 4, -3, -10, 9, -11}, {-5, -1, 6, 3, -2, 7}, {10, -9, 8, 5, -7, 2, -4, -6, 11, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 w u 3 + v w u 3 - v 2 w 2 u 2 + v w 2 u 2 + 2 v u 2 + 2 v 2 w u 2 -4 v w u 2 + w u 2 - u 2 + v 2 w 2 u -2 v w 2 u - v u - v 2 w u + 4 v w u -2 w u + u - v w + w$ (db)
Jones polynomial	$- q 2 + 3 q -5 + 8 q -1 -9 q -2 + 10 q -3 -8 q -4 + 7 q -5 -3 q -6 + 2 q -7$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a 8 z -2 + a 8 - z 4 a 6 -4 z 2 a 6 -2 a 6 z -2 -5 a 6 + z 6 a 4 + 4 z 4 a 4 + 6 z 2 a 4 + a 4 z -2 + 4 a 4 + z 6 a 2 + 3 z 4 a 2 + 2 z 2 a 2 - z 4 -2 z 2$ (db)
Kauffman polynomial	$a 5 z 9 + a 3 z 9 + 2 a 6 z 8 + 5 a 4 z 8 + 3 a 2 z 8 + a 7 z 7 + 3 a 3 z 7 + 4 a z 7 -7 a 6 z 6 -15 a 4 z 6 -5 a 2 z 6 + 3 z 6 - a 7 z 5 - a 5 z 5 -10 a 3 z 5 -9 a z 5 + z 5 a -1 + 3 a 8 z 4 + 20 a 6 z 4 + 27 a 4 z 4 + 3 a 2 z 4 -7 z 4 + 3 a 7 z 3 + 7 a 5 z 3 + 11 a 3 z 3 + 5 a z 3 -2 z 3 a -1 -8 a 8 z 2 -25 a 6 z 2 -22 a 4 z 2 -2 a 2 z 2 + 3 z 2 -6 a 7 z -8 a 5 z -3 a 3 z - a z + 6 a 8 + 12 a 6 + 8 a 4 + a 2 + 2 a 7 z -1 + 2 a 5 z -1 - a 8 z -2 -2 a 6 z -2 - a 4 z -2$ (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 L11n427. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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