L10n5

(Knotscape image)

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

Visit L10n5's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_17,1,18,4 X_9,14,10,15 X₃₈₄₉ X_5,11,6,10 X_13,5,14,20 X_11,19,12,18 X_19,13,20,12 X_2,16,3,15
Gauss code	{1, -10, -5, 3}, {-6, -1, 2, 5, -4, 6, -8, 9, -7, 4, 10, -2, -3, 8, -9, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + u 3 + v u 2 - u 2 - v u + u + v -1$ (db)
Jones polynomial	$-q^{11/2}+2 q^{9/2}-2 q^{7/2}+2 q^{5/2}-3 q^{3/2}+2 \sqrt{q}-\frac{2}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$- z 5 a -1 + a z 3 -5 z 3 a -1 + 2 z 3 a -3 + 3 a z -8 z a -1 + 6 z a -3 - z a -5 + 2 a z -1 -4 a -1 z -1 + 3 a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$- z 8 a -2 - z 8 - a z 7 -3 z 7 a -1 -2 z 7 a -3 + 3 z 6 a -2 -2 z 6 a -4 + 5 z 6 + 6 a z 5 + 16 z 5 a -1 + 9 z 5 a -3 - z 5 a -5 + 2 z 4 a -2 + 8 z 4 a -4 -6 z 4 -11 a z 3 -26 z 3 a -1 -13 z 3 a -3 + 2 z 3 a -5 -7 z 2 a -2 -9 z 2 a -4 -2 z 2 a -6 + 7 a z + 17 z a -1 + 11 z a -3 - z a -7 + 3 a -2 + 3 a -4 + a -6 + 2-2 a z -1 -4 a -1 z -1 -3 a -3 z -1 - a -5 z -1$ (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 L10n5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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