L11n180

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_3,10,4,11 X_5,14,6,15 X_16,8,17,7 X_22,18,7,17 X_15,13,16,12 X_9,20,10,21 X_11,19,12,18 X_13,6,14,1 X_19,4,20,5 X_2,21,3,22
Gauss code	{1, -11, -2, 10, -3, 9}, {4, -1, -7, 2, -8, 6, -9, 3, -6, -4, 5, 8, -10, 7, 11, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 2 + v u 2 + u 2 + v 2 u - v u + u + v 2 + v -1$ (db)
Jones polynomial	$-q^{5/2}+2 q^{3/2}-2 \sqrt{q}+\frac{2}{\sqrt{q}}-\frac{2}{q^{3/2}}+\frac{1}{q^{5/2}}-\frac{1}{q^{7/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a 7 z -1 - z a 5 - a 5 z -1 - z 3 a 3 -3 z a 3 + z 5 a + 4 z 3 a + 3 z a - z 3 a -1 -2 z a -1$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 - a 4 z 8 -3 a 2 z 8 -2 z 8 + 5 a 3 z 7 + 4 a z 7 - z 7 a -1 + 6 a 4 z 6 + 17 a 2 z 6 + 11 z 6 - a 7 z 5 -4 a 3 z 5 + 5 z 5 a -1 -8 a 4 z 4 -24 a 2 z 4 -16 z 4 + 5 a 7 z 3 + 3 a 5 z 3 -2 a 3 z 3 -6 a z 3 -6 z 3 a -1 + 2 a 6 z 2 + 3 a 4 z 2 + 8 a 2 z 2 + 7 z 2 -5 a 7 z -3 a 5 z + 2 a 3 z + 2 a z + 2 z a -1 - a 6 + a 7 z -1 + a 5 z -1$ (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 =

-3 is the signature of L11n180. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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