L10a40

(Knotscape image)

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

Visit L10a40's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_18,8,19,7 X_20,13,5,14 X_14,19,15,20 X_16,10,17,9 X_10,16,11,15 X_8,18,9,17 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -8, 6, -7, 10, -2, 4, -5, 7, -6, 8, -3, 5, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 u 3 -4 v u 2 + 7 u 2 + 7 v u -4 u -2 v$ (db)
Jones polynomial	$-q^{7/2}+2 q^{5/2}-4 q^{3/2}+6 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{9}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 7 z -1 -3 z a 5 - a 5 z -1 + 2 z 3 a 3 - a 3 z -1 + 3 z 3 a + 3 z a + 2 a z -1 + z 3 a -1 - z a -1 - a -1 z -1 - z a -3$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 -2 a 4 z 8 -5 a 2 z 8 -3 z 8 -2 a 5 z 7 - a 3 z 7 -2 a z 7 -3 z 7 a -1 -2 a 6 z 6 + 3 a 4 z 6 + 17 a 2 z 6 -2 z 6 a -2 + 10 z 6 - a 7 z 5 + a 5 z 5 + 7 a 3 z 5 + 15 a z 5 + 9 z 5 a -1 - z 5 a -3 + 4 a 6 z 4 -3 a 4 z 4 -26 a 2 z 4 + 5 z 4 a -2 -14 z 4 + 3 a 7 z 3 + 6 a 5 z 3 -13 a 3 z 3 -28 a z 3 -9 z 3 a -1 + 3 z 3 a -3 + 5 a 4 z 2 + 13 a 2 z 2 - z 2 a -2 + 7 z 2 -3 a 7 z -3 a 5 z + 8 a 3 z + 14 a z + 5 z a -1 - z a -3 - a 6 -2 a 4 -3 a 2 -1 + a 7 z -1 + a 5 z -1 - a 3 z -1 -2 a z -1 - a -1 z -1$ (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 L10a40. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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