L11n272

(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_12,21,13,22 X_20,13,21,14 X_14,19,15,20 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, 5, -7, 8, -9, -4, 3, 6, -5, 9, -8, 7, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 u 3 + 2 v u 2 + 2 w u 2 + u 2 -2 v u - v w u -2 w u + 2 v w$ (db)
Jones polynomial	$q 3 - q 2 + 2 q + 1 + q -1 + q -2 - q -3 + 2 q -4 -2 q -5 + q -6 - q -7$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 2 a 6 - a 6 z -2 -2 a 6 + z 4 a 4 + 4 z 2 a 4 + 3 a 4 z -2 + 6 a 4 - z 2 a 2 -2 a 2 z -2 -3 a 2 - z 4 -4 z 2 - z -2 -3 + z 2 a -2 + a -2 z -2 + 2 a -2$ (db)
Kauffman polynomial	$a 6 z 8 + a 4 z 8 + a 2 z 8 + z 8 + a 7 z 7 + 3 a 5 z 7 + 3 a 3 z 7 + 2 a z 7 + z 7 a -1 -5 a 6 z 6 -5 a 4 z 6 -6 a 2 z 6 + z 6 a -2 -5 z 6 -6 a 7 z 5 -18 a 5 z 5 -21 a 3 z 5 -13 a z 5 -4 z 5 a -1 + 6 a 6 z 4 + 5 a 4 z 4 + 7 a 2 z 4 -5 z 4 a -2 + 3 z 4 + 11 a 7 z 3 + 33 a 5 z 3 + 42 a 3 z 3 + 20 a z 3 -2 a 6 z 2 -4 a 4 z 2 -3 a 2 z 2 + 6 z 2 a -2 + 5 z 2 -7 a 7 z -27 a 5 z -34 a 3 z -10 a z + 4 z a -1 + a 6 + 5 a 4 + 4 a 2 -4 a -2 -3 + 2 a 7 z -1 + 8 a 5 z -1 + 10 a 3 z -1 + 2 a z -1 -2 a -1 z -1 - a 6 z -2 -3 a 4 z -2 -2 a 2 z -2 + a -2 z -2 + z -2$ (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 =

0 is the signature of L11n272. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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