L10n64

(Knotscape image)

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

Visit L10n64's page at the original Knot Atlas.

Planar diagram presentation	X_12,1,13,2 X_16,7,17,8 X₃₉₄₈ X_17,2,18,3 X_14,6,15,5 X_6,12,7,11 X_9,18,10,19 X_20,15,11,16 X_10,13,1,14 X_4,19,5,20
Gauss code	{1, 4, -3, -10, 5, -6, 2, 3, -7, -9}, {6, -1, 9, -5, 8, -2, -4, 7, 10, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{u^3 \left(-v^2\right)+2 u^3 v-u^3+u^2 v^2-3 u^2 v+u^2+u v^3-3 u v^2+u v-v^3+2 v^2-v}{u^{3/2} v^{3/2}}$ (db)
Jones polynomial	$\sqrt{q}-\frac{4}{\sqrt{q}}+\frac{5}{q^{3/2}}-\frac{6}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 7 (- z) + 2 a 5 z 3 + 3 a 5 z + a 5 z -1 - a 3 z 5 -3 a 3 z 3 -4 a 3 z - a 3 z -1 + a z 3$ (db)
Kauffman polynomial	$a 8 z 6 -3 a 8 z 4 + 2 a 8 z 2 + 3 a 7 z 7 -11 a 7 z 5 + 10 a 7 z 3 -2 a 7 z + 2 a 6 z 8 -4 a 6 z 6 -3 a 6 z 4 + 3 a 6 z 2 + 7 a 5 z 7 -24 a 5 z 5 + 23 a 5 z 3 -9 a 5 z + a 5 z -1 + 2 a 4 z 8 -3 a 4 z 6 -2 a 4 z 4 + 2 a 4 z 2 - a 4 + 4 a 3 z 7 -13 a 3 z 5 + 17 a 3 z 3 -8 a 3 z + a 3 z -1 + 2 a 2 z 6 -2 a 2 z 4 + 2 a 2 z 2 + 4 a z 3 - a z + z 2$ (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 L10n64. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -1$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 0$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = 1$	${\mathbb Z}_2$	${\mathbb Z}$