L11n279

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_2,14,3,13 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_15,1,16,4 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 $u$ , $v$ , $w$ , ...)	$v u 2 -2 v w u 2 + w u 2 + v 2 u -2 v u + 2 v w u - w u - v 2 + 2 v - v w$ (db)
Jones polynomial	$-2 q 6 + 3 q 5 -4 q 4 + 6 q 3 -4 q 2 + 5 q -2 + 2 q -1$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-2 z 4 a -2 -7 z 2 a -2 + 4 z 2 a -4 + 2 z 2 -10 a -2 + 8 a -4 -2 a -6 + 4-5 a -2 z -2 + 4 a -4 z -2 - a -6 z -2 + 2 z -2$ (db)
Kauffman polynomial	$z 8 a -2 + z 8 a -4 + z 7 a -1 + 3 z 7 a -3 + 2 z 7 a -5 -4 z 6 a -2 -3 z 6 a -4 + z 6 a -6 -3 z 5 a -1 -11 z 5 a -3 -8 z 5 a -5 + 13 z 4 a -2 + 10 z 4 a -4 + 3 z 4 + 7 z 3 a -1 + 24 z 3 a -3 + 20 z 3 a -5 + 3 z 3 a -7 -21 z 2 a -2 -14 z 2 a -4 -2 z 2 a -6 -9 z 2 -11 z a -1 -24 z a -3 -18 z a -5 -5 z a -7 + 15 a -2 + 12 a -4 + 3 a -6 + 7 + 5 a -1 z -1 + 9 a -3 z -1 + 5 a -5 z -1 + a -7 z -1 -5 a -2 z -2 -4 a -4 z -2 - a -6 z -2 -2 z -2$ (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 L11n279. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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