L10n63

(Knotscape image)

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

Visit L10n63's page at the original Knot Atlas.

Planar diagram presentation	X_12,1,13,2 X_7,17,8,16 X₃₉₄₈ X_17,2,18,3 X_5,14,6,15 X_11,6,12,7 X_9,18,10,19 X_15,11,16,20 X_10,13,1,14 X_19,5,20,4
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 $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + 2 v 2 u 3 - v u 3 + 2 v 3 u 2 -4 v 2 u 2 + 4 v u 2 - u 2 - v 3 u + 4 v 2 u -4 v u + 2 u - v 2 + 2 v -1$ (db)
Jones polynomial	$-q^{7/2}+4 q^{5/2}-7 q^{3/2}+9 \sqrt{q}-\frac{11}{\sqrt{q}}+\frac{10}{q^{3/2}}-\frac{9}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{3}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 - a 3 z 5 + 4 a z 5 - z 5 a -1 -3 a 3 z 3 + 4 a z 3 -2 z 3 a -1 + a 5 z -3 a 3 z + a 5 z -1 - a 3 z -1$ (db)
Kauffman polynomial	$-3 a 2 z 8 -3 z 8 -6 a 3 z 7 -12 a z 7 -6 z 7 a -1 -3 a 4 z 6 - a 2 z 6 -4 z 6 a -2 -2 z 6 + 13 a 3 z 5 + 26 a z 5 + 12 z 5 a -1 - z 5 a -3 + 6 a 2 z 4 + 7 z 4 a -2 + 13 z 4 -6 a 5 z 3 -18 a 3 z 3 -18 a z 3 -5 z 3 a -1 + z 3 a -3 -3 a 2 z 2 -2 z 2 a -2 -5 z 2 + 6 a 5 z + 9 a 3 z + 4 a z + z a -1 + a 4 - a 5 z -1 - a 3 z -1$ (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 =

-1 is the signature of L10n63. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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