L10n4

(Knotscape image)

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

Visit L10n4's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 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_15,2,16,3
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 $u$ , $v$ , $w$ , ...)	$- v u 5 + 2 v u 4 -2 v u 3 + 2 u 3 + 2 v u 2 -2 u 2 + 2 u -1$ (db)
Jones polynomial	$-q^{7/2}+2 q^{5/2}-3 q^{3/2}+4 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{5}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{2}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 - a 3 z 5 + 6 a z 5 - z 5 a -1 -5 a 3 z 3 + 12 a z 3 -4 z 3 a -1 + a 5 z -8 a 3 z + 9 a z -4 z a -1 + 2 a 5 z -1 -4 a 3 z -1 + 3 a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$- a 2 z 8 - z 8 -2 a 3 z 7 -4 a z 7 -2 z 7 a -1 - a 4 z 6 + 2 a 2 z 6 -2 z 6 a -2 + z 6 + 9 a 3 z 5 + 16 a z 5 + 6 z 5 a -1 - z 5 a -3 + 2 a 4 z 4 + 6 z 4 a -2 + 4 z 4 -3 a 5 z 3 -20 a 3 z 3 -25 a z 3 -5 z 3 a -1 + 3 z 3 a -3 -3 a 4 z 2 -7 a 2 z 2 -3 z 2 a -2 -7 z 2 + 7 a 5 z + 16 a 3 z + 13 a z + 3 z a -1 - z a -3 + 2 a 4 + 3 a 2 + a -2 + 3-2 a 5 z -1 -4 a 3 z -1 -3 a z -1 - a -1 z -1$ (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 L10n4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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