L10n109

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_2,16,3,15 X_16,7,17,8 X_19,11,20,14 X_13,15,14,20 X_9,18,10,19 X_11,10,12,5 X_17,1,18,4
Gauss code	{1, -4, -3, 10}, {-9, 2, -7, 6}, {-2, -1, 5, 3, -8, 9}, {4, -5, -10, 8, -6, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v x u 2 - v w x u 2 - x u 2 - v 2 u + 2 v u + v 2 w u - v w u - v x u + 2 v w x u - w x u + x u - v - v 2 w + v w$ (db)
Jones polynomial	$-2 q^{3/2}+2 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 3 a 5 + 2 z a 5 + 2 a 5 z -1 + a 5 z -3 - z 5 a 3 -4 z 3 a 3 -8 z a 3 -7 a 3 z -1 -3 a 3 z -3 + 3 z 3 a + 8 z a + 8 a z -1 + 3 a z -3 -2 z a -1 -3 a -1 z -1 - a -1 z -3$ (db)
Kauffman polynomial	$- a 4 z 8 - a 2 z 8 -3 a 5 z 7 -5 a 3 z 7 -2 a z 7 -2 a 6 z 6 - a 4 z 6 - z 6 - a 7 z 5 + 11 a 5 z 5 + 20 a 3 z 5 + 8 a z 5 + 5 a 6 z 4 + 11 a 4 z 4 + 8 a 2 z 4 + 2 z 4 + 3 a 7 z 3 -15 a 5 z 3 -36 a 3 z 3 -21 a z 3 -3 z 3 a -1 - a 6 z 2 -17 a 4 z 2 -23 a 2 z 2 -7 z 2 - a 7 z + 12 a 5 z + 29 a 3 z + 23 a z + 7 z a -1 + 10 a 4 + 19 a 2 + 10-5 a 5 z -1 -12 a 3 z -1 -12 a z -1 -5 a -1 z -1 -3 a 4 z -2 -6 a 2 z -2 -3 z -2 + a 5 z -3 + 3 a 3 z -3 + 3 a z -3 + a -1 z -3$ (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 =

-1 is the signature of L10n109. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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