L11n281

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₈₄₉₃ X_2,14,3,13 X_14,7,15,8 X_9,18,10,19 X_22,17,11,18 X_20,11,21,12 X_16,21,17,22 X_4,15,1,16 X_19,10,20,5
Gauss code	{1, -4, 3, -10}, {-2, -1, 5, -3, -6, 11}, {8, 2, 4, -5, 10, -9, 7, 6, -11, -8, 9, -7}

A Braid Representative

A Morse Link Presentation

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

-4 is the signature of L11n281. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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