L11n365

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,8,13,7 X_22,15,17,16 X_9,18,10,19 X_17,8,18,9 X_20,13,21,14 X_14,21,15,22 X_16,19,5,20 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {-6, 5, 9, -7, 8, -4}, {10, -1, 3, 6, -5, -2, 11, -3, 7, -8, 4, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 4 + v w u 4 + 2 v u 3 -2 v w u 3 + w u 3 - u 3 + v 2 u 2 -2 v u 2 - v 2 w u 2 + 2 v w u 2 - w u 2 + u 2 - v 2 u + 2 v u + v 2 w u -2 v w u - v + v w$ (db)
Jones polynomial	$-1 + 4 q -1 -5 q -2 + 8 q -3 -8 q -4 + 8 q -5 -6 q -6 + 5 q -7 -2 q -8 + q -9$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z 2 a 8 + a 8 z -2 + 2 a 8 -2 z 4 a 6 -6 z 2 a 6 -2 a 6 z -2 -6 a 6 + z 6 a 4 + 4 z 4 a 4 + 6 z 2 a 4 + a 4 z -2 + 3 a 4 - z 4 a 2 - z 2 a 2 + a 2$ (db)
Kauffman polynomial	$z 6 a 10 -4 z 4 a 10 + 5 z 2 a 10 -2 a 10 + 2 z 7 a 9 -6 z 5 a 9 + 3 z 3 a 9 + z a 9 + 2 z 8 a 8 -4 z 6 a 8 - z 4 a 8 - z 2 a 8 - a 8 z -2 + 3 a 8 + z 9 a 7 + z 7 a 7 -9 z 5 a 7 + 9 z 3 a 7 -7 z a 7 + 2 a 7 z -1 + 5 z 8 a 6 -17 z 6 a 6 + 25 z 4 a 6 -23 z 2 a 6 -2 a 6 z -2 + 11 a 6 + z 9 a 5 + z 7 a 5 -8 z 5 a 5 + 14 z 3 a 5 -9 z a 5 + 2 a 5 z -1 + 3 z 8 a 4 -12 z 6 a 4 + 26 z 4 a 4 -21 z 2 a 4 - a 4 z -2 + 7 a 4 + 2 z 7 a 3 -5 z 5 a 3 + 9 z 3 a 3 -2 z a 3 + 4 z 4 a 2 -4 z 2 a 2 + z 3 a - z a$ (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 =

-2 is the signature of L11n365. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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