L11n389

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 2 v u 4 -2 v w u 4 -2 v u 3 + 3 v w u 3 - w u 3 + u 3 + v u 2 - v w u 2 + 2 w u 2 -3 u 2 -2 w u + 2 u + w$ (db)
Jones polynomial	$q -3 -2 q -4 + 5 q -5 -6 q -6 + 9 q -7 -7 q -8 + 8 q -9 -6 q -10 + 3 q -11 - q -12$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$- a 14 z -2 + z 2 a 12 + 4 a 12 z -2 + 6 a 12 -3 z 4 a 10 -13 z 2 a 10 -5 a 10 z -2 -15 a 10 + 2 z 6 a 8 + 9 z 4 a 8 + 12 z 2 a 8 + 2 a 8 z -2 + 8 a 8 + z 6 a 6 + 4 z 4 a 6 + 4 z 2 a 6 + a 6$ (db)
Kauffman polynomial	$z 3 a 15 -2 z a 15 + a 15 z -1 + 3 z 4 a 14 -3 z 2 a 14 - a 14 z -2 + 2 a 14 + 2 z 7 a 13 -7 z 5 a 13 + 18 z 3 a 13 -15 z a 13 + 5 a 13 z -1 + 3 z 8 a 12 -12 z 6 a 12 + 26 z 4 a 12 -25 z 2 a 12 -4 a 12 z -2 + 14 a 12 + z 9 a 11 + 3 z 7 a 11 -22 z 5 a 11 + 44 z 3 a 11 -33 z a 11 + 9 a 11 z -1 + 6 z 8 a 10 -24 z 6 a 10 + 42 z 4 a 10 -42 z 2 a 10 -5 a 10 z -2 + 21 a 10 + z 9 a 9 + 3 z 7 a 9 -21 z 5 a 9 + 30 z 3 a 9 -20 z a 9 + 5 a 9 z -1 + 3 z 8 a 8 -11 z 6 a 8 + 15 z 4 a 8 -16 z 2 a 8 -2 a 8 z -2 + 9 a 8 + 2 z 7 a 7 -6 z 5 a 7 + 3 z 3 a 7 + z 6 a 6 -4 z 4 a 6 + 4 z 2 a 6 - a 6$ (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 =

-6 is the signature of L11n389. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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