L10n15

(Knotscape image)

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

Visit L10n15's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_4,19,1,20 X_14,10,15,9 X₈₄₉₃ X_5,13,6,12 X_13,5,14,20 X_16,12,17,11 X_10,16,11,15 X_2,18,3,17
Gauss code	{1, -10, 5, -3}, {-6, -1, 2, -5, 4, -9, 8, 6, -7, -4, 9, -8, 10, -2, 3, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 2 v u 4 -2 u 4 -2 v u 3 + 2 u 3 + 2 v u 2 -2 u 2 -2 v u + 2 u + v -1$ (db)
Jones polynomial	$2 q^{13/2}-4 q^{11/2}+6 q^{9/2}-7 q^{7/2}+6 q^{5/2}-7 q^{3/2}+4 \sqrt{q}-\frac{3}{\sqrt{q}}+\frac{1}{q^{3/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 - z 5 a -1 + 5 z 5 a -3 - z 5 a -5 -3 z 3 a -1 + 8 z 3 a -3 -4 z 3 a -5 - z a -1 + 5 z a -3 -5 z a -5 + z a -7 + a -1 z -1 -2 a -5 z -1 + a -7 z -1$ (db)
Kauffman polynomial	$-2 z 8 a -2 -2 z 8 a -4 -3 z 7 a -1 -8 z 7 a -3 -5 z 7 a -5 + 3 z 6 a -2 -4 z 6 a -6 - z 6 + 11 z 5 a -1 + 27 z 5 a -3 + 15 z 5 a -5 - z 5 a -7 + 6 z 4 a -2 + 12 z 4 a -4 + 9 z 4 a -6 + 3 z 4 -11 z 3 a -1 -26 z 3 a -3 -18 z 3 a -5 -3 z 3 a -7 -5 z 2 a -2 -12 z 2 a -4 -12 z 2 a -6 -3 z 2 a -8 -2 z 2 + 3 z a -1 + 10 z a -3 + 9 z a -5 + 2 z a -7 - a -2 + 3 a -4 + 5 a -6 + 2 a -8 + a -1 z -1 -2 a -5 z -1 - a -7 z -1$ (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 =

3 is the signature of L10n15. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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