L11n291

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 w u 4 + v w u 4 + v u 3 + 2 v 2 w u 3 -2 v w u 3 - u 3 -3 v u 2 -2 v 2 w u 2 + 3 v w u 2 + 2 u 2 + 2 v u + v 2 w u - v w u -2 u - v + 1$ (db)
Jones polynomial	$q 3 -3 q 2 + 5 q -7 + 9 q -1 -8 q -2 + 9 q -3 -5 q -4 + 4 q -5 - q -6$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- a 2 z 8 + a 4 z 6 -6 a 2 z 6 + z 6 + 5 a 4 z 4 -12 a 2 z 4 + 4 z 4 - a 6 z 2 + 7 a 4 z 2 -9 a 2 z 2 + 4 z 2 - a 6 + a 4 - a 2 + 1 + a 6 z -2 -2 a 4 z -2 + a 2 z -2$ (db)
Kauffman polynomial	$2 a 3 z 9 + 2 a z 9 + 4 a 4 z 8 + 8 a 2 z 8 + 4 z 8 + 2 a 5 z 7 -3 a 3 z 7 -2 a z 7 + 3 z 7 a -1 -17 a 4 z 6 -32 a 2 z 6 + z 6 a -2 -14 z 6 -5 a 5 z 5 -7 a 3 z 5 -12 a z 5 -10 z 5 a -1 + 4 a 6 z 4 + 32 a 4 z 4 + 44 a 2 z 4 -3 z 4 a -2 + 13 z 4 + a 7 z 3 + 9 a 5 z 3 + 19 a 3 z 3 + 18 a z 3 + 7 z 3 a -1 -5 a 6 z 2 -21 a 4 z 2 -24 a 2 z 2 + z 2 a -2 -7 z 2 - a 7 z -3 a 5 z -7 a 3 z -7 a z -2 z a -1 + 3 a 4 + 4 a 2 + 2-2 a 5 z -1 -2 a 3 z -1 + a 6 z -2 + 2 a 4 z -2 + a 2 z -2$ (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 L11n291. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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