L11n288

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_13,2,14,3 X_14,7,15,8 X_11,18,12,19 X_9,21,10,20 X_19,5,20,10 X_4,15,1,16 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 $u$ , $v$ , $w$ , ...)	$- v 2 w u 4 + v w u 4 + v 2 w u 3 - u 3 - v 2 w u 2 + u 2 + v 2 w u - u - v + 1$ (db)
Jones polynomial	$q -1 - q -2 + 3 q -3 -2 q -4 + 4 q -5 -2 q -6 + 3 q -7 -2 q -8 + q -9 - q -10$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$- z 2 a 10 - a 10 z -2 -2 a 10 + z 6 a 8 + 6 z 4 a 8 + 11 z 2 a 8 + 4 a 8 z -2 + 9 a 8 - z 8 a 6 -7 z 6 a 6 -17 z 4 a 6 -20 z 2 a 6 -5 a 6 z -2 -14 a 6 + z 6 a 4 + 6 z 4 a 4 + 11 z 2 a 4 + 2 a 4 z -2 + 7 a 4$ (db)
Kauffman polynomial	$z a 13 + z 2 a 12 + 2 z 3 a 11 -4 z a 11 + a 11 z -1 + z 6 a 10 -2 z 4 a 10 -4 z 2 a 10 - a 10 z -2 + 5 a 10 + 3 z 7 a 9 -16 z 5 a 9 + 27 z 3 a 9 -21 z a 9 + 5 a 9 z -1 + 3 z 8 a 8 -17 z 6 a 8 + 32 z 4 a 8 -32 z 2 a 8 -4 a 8 z -2 + 18 a 8 + z 9 a 7 -2 z 7 a 7 -12 z 5 a 7 + 34 z 3 a 7 -29 z a 7 + 9 a 7 z -1 + 4 z 8 a 6 -25 z 6 a 6 + 51 z 4 a 6 -45 z 2 a 6 -5 a 6 z -2 + 21 a 6 + z 9 a 5 -5 z 7 a 5 + 4 z 5 a 5 + 9 z 3 a 5 -13 z a 5 + 5 a 5 z -1 + z 8 a 4 -7 z 6 a 4 + 17 z 4 a 4 -18 z 2 a 4 -2 a 4 z -2 + 9 a 4$ (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 L11n288. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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