L10n103

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_14,7,15,8 X_8,13,5,14 X_11,18,12,19 X_20,16,17,15 X_16,20,9,19 X_17,12,18,13 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -4}, {-8, 5, 7, -6}, {10, -2, -5, 8, 4, -3, 6, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u + v w u - w u + v x u - v w x u + w x u - x u + u + v - v w + w - v x + v w x - w x + x -1$ (db)
Jones polynomial	$-\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{1}{q^{13/2}}-\frac{1}{q^{15/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a 9 z -3 -3 a 7 z -1 -3 a 7 z -3 - z 3 a 5 + 2 z a 5 + 6 a 5 z -1 + 3 a 5 z -3 + z 5 a 3 + 2 z 3 a 3 - z a 3 -3 a 3 z -1 - a 3 z -3 - z 3 a - z a$ (db)
Kauffman polynomial	$- z 3 a 9 + 3 z a 9 -3 a 9 z -1 + a 9 z -3 - z 4 a 8 -2 z 2 a 8 -3 a 8 z -2 + 6 a 8 - z 7 a 7 + 4 z 5 a 7 -12 z 3 a 7 + 9 z a 7 -6 a 7 z -1 + 3 a 7 z -3 - z 8 a 6 + 2 z 6 a 6 - z 4 a 6 -9 z 2 a 6 -6 a 6 z -2 + 11 a 6 -4 z 7 a 5 + 12 z 5 a 5 -16 z 3 a 5 + 11 z a 5 -6 a 5 z -1 + 3 a 5 z -3 - z 8 a 4 - z 6 a 4 + 7 z 4 a 4 -8 z 2 a 4 -3 a 4 z -2 + 6 a 4 -3 z 7 a 3 + 7 z 5 a 3 -3 z 3 a 3 + 4 z a 3 -3 a 3 z -1 + a 3 z -3 -3 z 6 a 2 + 7 z 4 a 2 - z 2 a 2 - z 5 a + 2 z 3 a - z a$ (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 L10n103. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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