L11n270

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_16,7,17,8 X_8,15,5,16 X_11,19,12,18 X_17,9,18,22 X_13,21,14,20 X_19,13,20,12 X_21,15,22,14 X₂₅₃₆ X_4,9,1,10
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 $u$ , $v$ , $w$ , ...)	$- v u 5 - w u 5 + u 5 + v u 4 + w u 4 - v u 3 - w u 3 + v u 2 + w u 2 - v u - w u + v - v w + w$ (db)
Jones polynomial	$-2 q 4 + 2 q 3 -3 q 2 + 4 q -4 + 5 q -1 -2 q -2 + 4 q -3 - q -4 + q -5$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$2 z 6 -3 a 2 z 4 -2 z 4 a -2 + 12 z 4 + a 4 z 2 -14 a 2 z 2 -8 z 2 a -2 + 24 z 2 + 4 a 4 -18 a 2 -7 a -2 + 21 + 3 a 4 z -2 -8 a 2 z -2 -2 a -2 z -2 + 7 z -2$ (db)
Kauffman polynomial	$a 3 z 9 + a z 9 + a 4 z 8 + 5 a 2 z 8 + 4 z 8 -4 a 3 z 7 + 4 z 7 a -1 -7 a 4 z 6 -30 a 2 z 6 + 2 z 6 a -2 -21 z 6 -2 a 3 z 5 -21 a z 5 -17 z 5 a -1 + 2 z 5 a -3 + 18 a 4 z 4 + 59 a 2 z 4 -2 z 4 a -2 + z 4 a -4 + 38 z 4 + 22 a 3 z 3 + 50 a z 3 + 24 z 3 a -1 -4 z 3 a -3 -22 a 4 z 2 -51 a 2 z 2 -8 z 2 a -2 -37 z 2 -24 a 3 z -45 a z -21 z a -1 + 3 z a -3 + 3 z a -5 + 13 a 4 + 28 a 2 + 7 a -2 + a -4 + 22 + 8 a 3 z -1 + 15 a z -1 + 7 a -1 z -1 - a -3 z -1 - a -5 z -1 -3 a 4 z -2 -8 a 2 z -2 -2 a -2 z -2 -7 z -2$ (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 L11n270. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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