L11n59

(Knotscape image)

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

Visit L11n59's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_14,8,15,7 X_11,19,12,18 X_19,5,20,22 X_15,21,16,20 X_21,17,22,16 X_17,13,18,12 X_8,14,9,13 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, -4, 8, 9, -3, -6, 7, -8, 4, -5, 6, -7, 5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$u 5 - v u 4 + u 4 + 4 v u 3 -4 u 3 -4 v u 2 + 4 u 2 + v u - u + v$ (db)
Jones polynomial	$q^{15/2}-3 q^{13/2}+5 q^{11/2}-6 q^{9/2}+7 q^{7/2}-6 q^{5/2}+5 q^{3/2}-4 \sqrt{q}-\frac{1}{q^{5/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 5 a -1 + z 5 a -3 + a z 3 -7 z 3 a -1 + 3 z 3 a -3 -2 z 3 a -5 + 4 a z -10 z a -1 + 5 z a -3 -2 z a -5 + z a -7 + 3 a z -1 -5 a -1 z -1 + 2 a -3 z -1$ (db)
Kauffman polynomial	$- z 9 a -3 - z 9 a -5 - z 8 a -2 -4 z 8 a -4 -3 z 8 a -6 - a z 7 + 3 z 7 a -3 - z 7 a -5 -3 z 7 a -7 + 5 z 6 a -2 + 15 z 6 a -4 + 9 z 6 a -6 - z 6 a -8 + 7 a z 5 + 4 z 5 a -1 -2 z 5 a -3 + 11 z 5 a -5 + 10 z 5 a -7 -7 z 4 a -2 -19 z 4 a -4 -5 z 4 a -6 + 3 z 4 a -8 + 4 z 4 -14 a z 3 -16 z 3 a -1 -3 z 3 a -3 -8 z 3 a -5 -7 z 3 a -7 -5 z 2 a -2 + 10 z 2 a -4 + 3 z 2 a -6 -2 z 2 a -8 -10 z 2 + 11 a z + 14 z a -1 + 3 z a -3 + z a -5 + z a -7 + 5 a -2 - a -6 + 5-3 a z -1 -5 a -1 z -1 -2 a -3 z -1$ (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 L11n59. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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