L11n159

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_7,17,8,16 X_10,4,11,3 X_2,15,3,16 X_14,10,15,9 X_11,19,12,18 X_5,13,6,12 X_21,1,22,6 X_20,14,21,13 X_17,7,18,22 X_4,20,5,19
Gauss code	{1, -4, 3, -11, -7, 8}, {-2, -1, 5, -3, -6, 7, 9, -5, 4, 2, -10, 6, 11, -9, -8, 10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{u^2 v^4-2 u^2 v^3+u^2 v^2-u v^4+u v^2-u+v^2-2 v+1}{u v^2}$ (db)
Jones polynomial	$2 q^{15/2}-2 q^{13/2}+2 q^{11/2}-4 q^{9/2}+2 q^{7/2}-3 q^{5/2}+2 q^{3/2}-\sqrt{q}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$- z 7 a -5 + z 5 a -3 -5 z 5 a -5 + z 5 a -7 + 4 z 3 a -3 -6 z 3 a -5 + 4 z 3 a -7 + 3 z a -3 -2 z a -5 + z a -7 - z a -9 + a -3 z -1 -2 a -7 z -1 + a -9 z -1$ (db)
Kauffman polynomial	$- z 9 a -5 - z 9 a -7 -2 z 8 a -4 -3 z 8 a -6 - z 8 a -8 - z 7 a -3 + 3 z 7 a -5 + 4 z 7 a -7 + 10 z 6 a -4 + 15 z 6 a -6 + 5 z 6 a -8 + 5 z 5 a -3 + 4 z 5 a -5 - z 5 a -7 -12 z 4 a -4 -19 z 4 a -6 -7 z 4 a -8 -7 z 3 a -3 -12 z 3 a -5 -5 z 3 a -7 + 2 z 2 a -4 + 8 z 2 a -6 + 7 z 2 a -8 + z 2 a -10 + 4 z a -3 + 6 z a -5 + 3 z a -7 + z a -9 + a -4 -3 a -6 -5 a -8 -2 a -10 - a -3 z -1 + 2 a -7 z -1 + a -9 z -1$ (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 =

5 is the signature of L11n159. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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