L10n41

(Knotscape image)

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

Visit L10n41's page at the original Knot Atlas.

Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_20,10,7,9 X₂₇₃₈ X_4,15,5,16 X_5,13,6,12 X_11,16,12,17 X_17,6,18,1 X_14,19,15,20 X_18,13,19,14
Gauss code	{1, -4, 2, -5, -6, 8}, {4, -1, 3, -2, -7, 6, 10, -9, 5, 7, -8, -10, 9, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 2 + 2 v u 2 - u 2 + v 2 u -3 v u + u - v 2 + 2 v -1$ (db)
Jones polynomial	$\sqrt{q}-\frac{3}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{5}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{2}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 - a 7 z -1 + 2 z 3 a 5 + 5 z a 5 + 3 a 5 z -1 - z 5 a 3 -4 z 3 a 3 -6 z a 3 -2 a 3 z -1 + z 3 a + z a$ (db)
Kauffman polynomial	$- z 6 a 8 + 4 z 4 a 8 -4 z 2 a 8 + a 8 -2 z 7 a 7 + 8 z 5 a 7 -8 z 3 a 7 + 3 z a 7 - a 7 z -1 - z 8 a 6 + z 6 a 6 + 7 z 4 a 6 -9 z 2 a 6 + 3 a 6 -4 z 7 a 5 + 15 z 5 a 5 -17 z 3 a 5 + 10 z a 5 -3 a 5 z -1 - z 8 a 4 + z 6 a 4 + 4 z 4 a 4 -6 z 2 a 4 + 3 a 4 -2 z 7 a 3 + 7 z 5 a 3 -12 z 3 a 3 + 9 z a 3 -2 a 3 z -1 - z 6 a 2 + z 4 a 2 -2 z 2 a 2 -3 z 3 a + 2 z a - z 2$ (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 L10n41. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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