L11n360

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_5,18,6,19 X₈₄₉₃ X_9,21,10,20 X_19,11,20,10 X_17,14,18,15 X_15,22,16,17 X_21,16,22,5 X_2,12,3,11
Gauss code	{1, -11, 5, -3}, {-8, 4, -7, 6, -10, 9}, {-4, -1, 2, -5, -6, 7, 11, -2, 3, 8, -9, 10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 2 + u 2 + 2 v u -2 u - v + 1$ (db)
Jones polynomial	$q 3 -2 q 2 + 3 q -3 + 4 q -1 -2 q -2 + 3 q -3 + q -6 - q -7$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- z 2 a 6 -2 a 6 + z 4 a 4 + 5 z 2 a 4 + a 4 z -2 + 5 a 4 - z 4 a 2 -3 z 2 a 2 -2 a 2 z -2 -4 a 2 - z 4 -2 z 2 + z -2 + z 2 a -2 + a -2$ (db)
Kauffman polynomial	$a 3 z 9 + a z 9 + a 6 z 8 + 2 a 4 z 8 + 3 a 2 z 8 + 2 z 8 + a 7 z 7 + a 5 z 7 -5 a 3 z 7 -3 a z 7 + 2 z 7 a -1 -7 a 6 z 6 -16 a 4 z 6 -18 a 2 z 6 + z 6 a -2 -8 z 6 -6 a 7 z 5 -9 a 5 z 5 + 2 a 3 z 5 -3 a z 5 -8 z 5 a -1 + 14 a 6 z 4 + 39 a 4 z 4 + 35 a 2 z 4 -4 z 4 a -2 + 6 z 4 + 9 a 7 z 3 + 18 a 5 z 3 + 13 a 3 z 3 + 10 a z 3 + 6 z 3 a -1 -12 a 6 z 2 -38 a 4 z 2 -31 a 2 z 2 + 3 z 2 a -2 -2 z 2 -3 a 7 z -10 a 5 z -13 a 3 z -7 a z - z a -1 + 5 a 6 + 15 a 4 + 13 a 2 - a -2 + 3 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 L11n360. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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