L10n10

(Knotscape image)

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

Visit L10n10's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_5,12,6,13 X₃₈₄₉ X_9,16,10,17 X_17,20,18,5 X_11,19,12,18 X_19,11,20,10 X_13,2,14,3
Gauss code	{1, 10, -5, -3}, {-4, -1, 2, 5, -6, 9, -8, 4, -10, -2, 3, 6, -7, 8, -9, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 2 v u 4 + 2 u -1$ (db)
Jones polynomial	$-\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{2}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{2}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z a 9 + a 9 z -1 - z 5 a 7 -5 z 3 a 7 -6 z a 7 -3 a 7 z -1 + z 7 a 5 + 6 z 5 a 5 + 11 z 3 a 5 + 9 z a 5 + 4 a 5 z -1 - z 5 a 3 -5 z 3 a 3 -6 z a 3 -2 a 3 z -1$ (db)
Kauffman polynomial	$- z a 11 - z 2 a 10 + a 10 - z 3 a 9 + 2 z a 9 - a 9 z -1 - z 6 a 8 + 4 z 4 a 8 -4 z 2 a 8 + 3 a 8 -2 z 7 a 7 + 11 z 5 a 7 -18 z 3 a 7 + 12 z a 7 -3 a 7 z -1 - z 8 a 6 + 4 z 6 a 6 - z 4 a 6 -5 z 2 a 6 + 3 a 6 -3 z 7 a 5 + 17 z 5 a 5 -28 z 3 a 5 + 17 z a 5 -4 a 5 z -1 - z 8 a 4 + 5 z 6 a 4 -5 z 4 a 4 -2 z 2 a 4 + 2 a 4 - z 7 a 3 + 6 z 5 a 3 -11 z 3 a 3 + 8 z a 3 -2 a 3 z -1$ (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 =

-5 is the signature of L10n10. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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