L11n320

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_5,14,6,15 X₃₈₄₉ X_15,2,16,3 X_16,7,17,8 X_9,18,10,19 X_4,17,1,18 X_19,13,20,22 X_13,10,14,11 X_21,5,22,12 X_11,21,12,20
Gauss code	{1, 4, -3, -7}, {-2, -1, 5, 3, -6, 9, -11, 10}, {-9, 2, -4, -5, 7, 6, -8, 11, -10, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 w u 3 + v 2 w u 3 + v 3 w u 2 -2 v 2 w u 2 + 2 v u - u - v + 1$ (db)
Jones polynomial	$- q + 2-2 q -1 + 3 q -2 -2 q -3 + 4 q -4 -2 q -5 + 2 q -6 - q -7 + q -8$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$z 2 a 8 + a 8 z -2 + 2 a 8 - z 6 a 6 -6 z 4 a 6 -11 z 2 a 6 -2 a 6 z -2 -9 a 6 + z 8 a 4 + 7 z 6 a 4 + 17 z 4 a 4 + 19 z 2 a 4 + a 4 z -2 + 9 a 4 - z 6 a 2 -5 z 4 a 2 -6 z 2 a 2 -2 a 2$ (db)
Kauffman polynomial	$z 2 a 10 - a 10 + z 3 a 9 - z a 9 + z 4 a 8 -2 z 2 a 8 - a 8 z -2 + 3 a 8 + z 7 a 7 -5 z 5 a 7 + 9 z 3 a 7 -8 z a 7 + 2 a 7 z -1 + 2 z 8 a 6 -12 z 6 a 6 + 25 z 4 a 6 -25 z 2 a 6 -2 a 6 z -2 + 11 a 6 + z 9 a 5 -3 z 7 a 5 -5 z 5 a 5 + 17 z 3 a 5 -12 z a 5 + 2 a 5 z -1 + 4 z 8 a 4 -23 z 6 a 4 + 41 z 4 a 4 -32 z 2 a 4 - a 4 z -2 + 11 a 4 + z 9 a 3 -3 z 7 a 3 -5 z 5 a 3 + 15 z 3 a 3 -7 z a 3 + 2 z 8 a 2 -11 z 6 a 2 + 17 z 4 a 2 -10 z 2 a 2 + 3 a 2 + z 7 a -5 z 5 a + 6 z 3 a -2 z a$ (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 =

-4 is the signature of L11n320. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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