L10n111

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_13,17,14,20 X_19,11,20,16 X_7,19,8,18 X_15,8,16,9 X_9,14,10,15 X_17,5,18,10 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -9, 2, -10}, {-8, 5, -4, 3}, {9, -1, -5, 6, -7, 8}, {10, -2, -3, 7, -6, 4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 w u 2 + v w u 2 - v x u 2 + v 2 w u + v w u - w u - v 2 x u + v x u + x u - v w + v x - x$ (db)
Jones polynomial	$-q^{7/2}+q^{5/2}-q^{3/2}-\frac{2}{\sqrt{q}}-\frac{2}{q^{3/2}}-\frac{1}{q^{5/2}}-\frac{2}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z a 5 + 2 a 5 z -1 + a 5 z -3 -2 z 3 a 3 -8 z a 3 -7 a 3 z -1 -3 a 3 z -3 + z 5 a + 6 z 3 a + 10 z a + 8 a z -1 + 3 a z -3 -2 z a -1 -3 a -1 z -1 - a -1 z -3 - z a -3$ (db)
Kauffman polynomial	$- a 4 z 8 - a 2 z 8 - a 5 z 7 -3 a 3 z 7 -3 a z 7 - z 7 a -1 + 5 a 4 z 6 + 4 a 2 z 6 - z 6 a -2 -2 z 6 + 6 a 5 z 5 + 20 a 3 z 5 + 21 a z 5 + 6 z 5 a -1 - z 5 a -3 -3 a 4 z 4 + 6 a 2 z 4 + 4 z 4 a -2 + 13 z 4 -11 a 5 z 3 -39 a 3 z 3 -43 a z 3 -11 z 3 a -1 + 4 z 3 a -3 -8 a 4 z 2 -26 a 2 z 2 -2 z 2 a -2 -20 z 2 + 10 a 5 z + 31 a 3 z + 35 a z + 12 z a -1 -2 z a -3 + 10 a 4 + 19 a 2 + 10-5 a 5 z -1 -12 a 3 z -1 -12 a z -1 -5 a -1 z -1 -3 a 4 z -2 -6 a 2 z -2 -3 z -2 + a 5 z -3 + 3 a 3 z -3 + 3 a z -3 + a -1 z -3$ (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 L10n111. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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