L11n361

(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_20,9,21,10 X_10,19,11,20 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 $u$ , $v$ , $w$ , ...)	$- w u 4 + u 4 + v u 3 - v w u 3 + 2 w u 3 -2 u 3 + v 2 u 2 -2 v u 2 - v 2 w u 2 + 2 v w u 2 - w u 2 + u 2 -2 v 2 u + v u + 2 v 2 w u - v w u + v 2 - v 2 w$ (db)
Jones polynomial	$2 q -1 -3 q -2 + 6 q -3 -7 q -4 + 8 q -5 -7 q -6 + 7 q -7 -4 q -8 + 3 q -9 - q -10$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- a 10 + 3 z 2 a 8 + a 8 z -2 + 4 a 8 -2 z 4 a 6 -5 z 2 a 6 -2 a 6 z -2 -6 a 6 - z 4 a 4 + a 4 z -2 + a 4 + 2 z 2 a 2 + 2 a 2$ (db)
Kauffman polynomial	$z 7 a 11 -4 z 5 a 11 + 4 z 3 a 11 - z a 11 + 3 z 8 a 10 -14 z 6 a 10 + 19 z 4 a 10 -8 z 2 a 10 + 2 z 9 a 9 -5 z 7 a 9 -5 z 5 a 9 + 12 z 3 a 9 -2 z a 9 + 8 z 8 a 8 -35 z 6 a 8 + 47 z 4 a 8 -27 z 2 a 8 - a 8 z -2 + 7 a 8 + 2 z 9 a 7 -2 z 7 a 7 -12 z 5 a 7 + 17 z 3 a 7 -9 z a 7 + 2 a 7 z -1 + 5 z 8 a 6 -19 z 6 a 6 + 28 z 4 a 6 -25 z 2 a 6 -2 a 6 z -2 + 11 a 6 + 4 z 7 a 5 -10 z 5 a 5 + 10 z 3 a 5 -7 z a 5 + 2 a 5 z -1 + 2 z 6 a 4 -3 z 2 a 4 - a 4 z -2 + 3 a 4 + z 5 a 3 + z 3 a 3 + z a 3 + 3 z 2 a 2 -2 a 2$ (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 L11n361. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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