L10n25

(Knotscape image)

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

Visit L10n25's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,17,12,16 X_7,15,8,14 X_15,9,16,8 X_17,5,18,20 X_13,18,14,19 X_19,12,20,13 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 $u$ , $v$ , $w$ , ...)	$- u 5 + 2 u 4 - v u 3 + u 3 + v u 2 - u 2 + 2 v u - v$ (db)
Jones polynomial	$-q^{7/2}+q^{5/2}-\frac{1}{q^{3/2}}+\frac{1}{q^{5/2}}-\frac{2}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z a 5 + 2 a 5 z -1 -2 z 3 a 3 -7 z a 3 -4 a 3 z -1 + z 5 a + 5 z 3 a + 6 z a + 3 a z -1 - z a -1 - a -1 z -1 - z a -3$ (db)
Kauffman polynomial	$- a 4 z 8 - a 2 z 8 - a 5 z 7 -3 a 3 z 7 -2 a z 7 + 5 a 4 z 6 + 5 a 2 z 6 - z 6 a -2 - z 6 + 6 a 5 z 5 + 18 a 3 z 5 + 13 a z 5 - z 5 a -3 -5 a 4 z 4 -3 a 2 z 4 + 5 z 4 a -2 + 7 z 4 -11 a 5 z 3 -30 a 3 z 3 -22 a z 3 + z 3 a -1 + 4 z 3 a -3 - a 4 z 2 -5 a 2 z 2 -5 z 2 a -2 -9 z 2 + 8 a 5 z + 18 a 3 z + 13 a z + z a -1 -2 z a -3 + 2 a 4 + 3 a 2 + a -2 + 3-2 a 5 z -1 -4 a 3 z -1 -3 a z -1 - a -1 z -1$ (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 L10n25. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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