L10n26

(Knotscape image)

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

Visit L10n26's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_16,11,17,12 X_7,15,8,14 X_15,9,16,8 X_20,17,5,18 X_18,14,19,13 X_12,20,13,19 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, -4, 5, 10, -2, 3, -8, 7, 4, -5, -3, 6, -7, 8, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + u 3 + 4 v u 2 -5 u 2 -5 v u + 4 u + v -1$ (db)
Jones polynomial	$-q^{7/2}+3 q^{5/2}-5 q^{3/2}+7 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{7}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{2}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 5 z -1 + z 3 a 3 - z a 3 - a 3 z -1 - z 5 a -2 z 3 a -2 z a + 2 z 3 a -1 + 2 z a -1 - z a -3$ (db)
Kauffman polynomial	$- a 2 z 8 - z 8 -2 a 3 z 7 -5 a z 7 -3 z 7 a -1 - a 4 z 6 -3 a 2 z 6 -3 z 6 a -2 -5 z 6 + a 3 z 5 + 6 a z 5 + 4 z 5 a -1 - z 5 a -3 -2 a 4 z 4 + 4 a 2 z 4 + 7 z 4 a -2 + 13 z 4 -3 a 5 z 3 -3 a 3 z 3 + 2 z 3 a -1 + 2 z 3 a -3 - a 2 z 2 -4 z 2 a -2 -5 z 2 + 3 a 5 z + 3 a 3 z - z a -1 - z a -3 + a 4 - a 5 z -1 - a 3 z -1$ (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 L10n26. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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