L11n269

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_7,17,8,16 X_15,5,16,8 X_18,11,19,12 X_22,17,9,18 X_20,13,21,14 X_12,19,13,20 X_14,21,15,22 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, 5, -8, 7, -9, -4, 3, 6, -5, 8, -7, 9, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$u 5 - v u 4 - w u 4 + v u 3 + w u 3 - v u 2 - w u 2 + v u + w u - v w$ (db)
Jones polynomial	$q 2 + 1 + 2 q -1 + 3 q -3 -3 q -4 + 2 q -5 -2 q -6 + q -7 - q -8$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- z 4 a 6 -4 z 2 a 6 -2 a 6 z -2 -5 a 6 + z 6 a 4 + 7 z 4 a 4 + 18 z 2 a 4 + 7 a 4 z -2 + 18 a 4 - z 6 a 2 -8 z 4 a 2 -20 z 2 a 2 -8 a 2 z -2 -20 a 2 + z 4 + 5 z 2 + 3 z -2 + 7$ (db)
Kauffman polynomial	$z 5 a 9 -4 z 3 a 9 + 3 z a 9 - a 9 z -1 + z 6 a 8 -3 z 4 a 8 + a 8 + z 7 a 7 -3 z 5 a 7 + 3 z a 7 - a 7 z -1 + z 8 a 6 -5 z 6 a 6 + 9 z 4 a 6 -9 z 2 a 6 -2 a 6 z -2 + 7 a 6 + 2 z 7 a 5 -12 z 5 a 5 + 26 z 3 a 5 -21 z a 5 + 7 a 5 z -1 + z 8 a 4 -8 z 6 a 4 + 24 z 4 a 4 -32 z 2 a 4 -7 a 4 z -2 + 22 a 4 + 2 z 7 a 3 -17 z 5 a 3 + 46 z 3 a 3 -45 z a 3 + 15 a 3 z -1 + z 8 a 2 -10 z 6 a 2 + 33 z 4 a 2 -46 z 2 a 2 -8 a 2 z -2 + 28 a 2 + z 7 a -9 z 5 a + 24 z 3 a -24 z a + 8 a z -1 + z 8 -8 z 6 + 21 z 4 -23 z 2 -3 z -2 + 13$ (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 L11n269. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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