L11n289

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_2,14,3,13 X_14,7,15,8 X_11,18,12,19 X_9,21,10,20 X_19,5,20,10 X_15,1,16,4 X_17,22,18,11 X_21,16,22,17
Gauss code	{1, -4, -3, 9}, {-2, -1, 5, 3, -7, 8}, {-6, 2, 4, -5, -9, 11, -10, 6, -8, 7, -11, 10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v w u 4 + w u 4 + v 2 u 3 -2 v u 3 + 2 v w u 3 - w u 3 - v 2 u 2 + 2 v u 2 -2 v w u 2 + w u 2 + v 2 u -2 v u + 2 v w u - w u - v 2 + v$ (db)
Jones polynomial	$2 q -2 + 6 q -1 -6 q -2 + 8 q -3 -7 q -4 + 6 q -5 -4 q -6 + 2 q -7 - q -8$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- z 4 a 6 -3 z 2 a 6 - a 6 z -2 -3 a 6 + z 6 a 4 + 5 z 4 a 4 + 11 z 2 a 4 + 4 a 4 z -2 + 11 a 4 -3 z 4 a 2 -11 z 2 a 2 -5 a 2 z -2 -13 a 2 + 2 z 2 + 2 z -2 + 5$ (db)
Kauffman polynomial	$z 5 a 9 -3 z 3 a 9 + z a 9 + 2 z 6 a 8 -5 z 4 a 8 + z 2 a 8 + 3 z 7 a 7 -9 z 5 a 7 + 8 z 3 a 7 -4 z a 7 + a 7 z -1 + 3 z 8 a 6 -11 z 6 a 6 + 17 z 4 a 6 -12 z 2 a 6 - a 6 z -2 + 5 a 6 + z 9 a 5 + z 7 a 5 -14 z 5 a 5 + 31 z 3 a 5 -21 z a 5 + 5 a 5 z -1 + 5 z 8 a 4 -23 z 6 a 4 + 47 z 4 a 4 -40 z 2 a 4 -4 a 4 z -2 + 18 a 4 + z 9 a 3 - z 7 a 3 -7 z 5 a 3 + 28 z 3 a 3 -29 z a 3 + 9 a 3 z -1 + 2 z 8 a 2 -10 z 6 a 2 + 28 z 4 a 2 -37 z 2 a 2 -5 a 2 z -2 + 21 a 2 + z 7 a -3 z 5 a + 8 z 3 a -13 z a + 5 a z -1 + 3 z 4 -10 z 2 -2 z -2 + 9$ (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 L11n289. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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