L10n108

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_15,2,16,3 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_4,17,1,18
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 $u$ , $v$ , $w$ , ...)	$v 2 w u 2 - v 2 w x u 2 + v w x u 2 - v 2 w u - x u + v + x -1$ (db)
Jones polynomial	$-\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{1}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z a 9 + a 9 z -1 + a 9 z -3 - z 5 a 7 -5 z 3 a 7 -7 z a 7 -6 a 7 z -1 -3 a 7 z -3 + z 7 a 5 + 6 z 5 a 5 + 12 z 3 a 5 + 13 z a 5 + 9 a 5 z -1 + 3 a 5 z -3 - z 5 a 3 -5 z 3 a 3 -7 z a 3 -4 a 3 z -1 - a 3 z -3$ (db)
Kauffman polynomial	$- z a 11 - z 2 a 10 -3 z 3 a 9 + 8 z a 9 -5 a 9 z -1 + a 9 z -3 -2 z 6 a 8 + 9 z 4 a 8 -13 z 2 a 8 -3 a 8 z -2 + 10 a 8 -3 z 7 a 7 + 16 z 5 a 7 -28 z 3 a 7 + 25 z a 7 -12 a 7 z -1 + 3 a 7 z -3 - z 8 a 6 + 2 z 6 a 6 + 9 z 4 a 6 -23 z 2 a 6 -6 a 6 z -2 + 19 a 6 -4 z 7 a 5 + 22 z 5 a 5 -37 z 3 a 5 + 27 z a 5 -12 a 5 z -1 + 3 a 5 z -3 - z 8 a 4 + 4 z 6 a 4 -11 z 2 a 4 -3 a 4 z -2 + 10 a 4 - z 7 a 3 + 6 z 5 a 3 -12 z 3 a 3 + 11 z a 3 -5 a 3 z -1 + a 3 z -3$ (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 =

-5 is the signature of L10n108. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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