L11n278

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_13,2,14,3 X_14,7,15,8 X_18,10,19,9 X_17,11,18,22 X_11,21,12,20 X_21,17,22,16 X_4,15,1,16 X_10,20,5,19
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 $u$ , $v$ , $w$ , ...)	$- v 2 w u 2 + v w u 2 + v 2 u - w u - v + 1$ (db)
Jones polynomial	$- q 4 + q 3 - q 2 + q + q -1 + q -2 + q -3 + q -4$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z 6 - a 2 z 4 - z 4 a -2 + 6 z 4 -6 a 2 z 2 -4 z 2 a -2 + 11 z 2 + 2 a 4 -9 a 2 -3 a -2 + 10 + 2 a 4 z -2 -5 a 2 z -2 - a -2 z -2 + 4 z -2$ (db)
Kauffman polynomial	$z 8 a -2 + z 8 + a z 7 + 2 z 7 a -1 + z 7 a -3 - a 2 z 6 -6 z 6 a -2 -7 z 6 - a 3 z 5 -8 a z 5 -13 z 5 a -1 -6 z 5 a -3 + a 4 z 4 + 7 a 2 z 4 + 10 z 4 a -2 + 16 z 4 + 6 a 3 z 3 + 21 a z 3 + 25 z 3 a -1 + 10 z 3 a -3 -5 a 4 z 2 -17 a 2 z 2 -6 z 2 a -2 -18 z 2 -11 a 3 z -24 a z -18 z a -1 -5 z a -3 + 7 a 4 + 15 a 2 + 3 a -2 + 12 + 5 a 3 z -1 + 9 a z -1 + 5 a -1 z -1 + a -3 z -1 -2 a 4 z -2 -5 a 2 z -2 - a -2 z -2 -4 z -2$ (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 L11n278. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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