L11n87

(Knotscape image)

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

Visit L11n87's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_14,8,15,7 X_15,20,16,21 X_9,19,10,18 X_19,9,20,8 X_17,22,18,5 X_21,16,22,17 X_10,14,11,13 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, 6, -5, -9, 11, -2, 9, -3, -4, 8, -7, 5, -6, 4, -8, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + u 3 + v u 2 + u + v -1$ (db)
Jones polynomial	$-q^{5/2}+2 q^{3/2}-2 \sqrt{q}+\frac{2}{\sqrt{q}}-\frac{2}{q^{3/2}}+\frac{1}{q^{5/2}}-\frac{2}{q^{7/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a 5 z -1 - z 3 a 3 -3 z a 3 - a 3 z -1 + z 5 a + 4 z 3 a + 3 z a - z 3 a -1 -2 z a -1$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 - a 4 z 8 -3 a 2 z 8 -2 z 8 + 5 a 3 z 7 + 4 a z 7 - z 7 a -1 + 6 a 4 z 6 + 17 a 2 z 6 + 11 z 6 -5 a 3 z 5 + 5 z 5 a -1 -9 a 4 z 4 -25 a 2 z 4 -16 z 4 + a 5 z 3 + a 3 z 3 -6 a z 3 -6 z 3 a -1 + a 6 z 2 + 4 a 4 z 2 + 10 a 2 z 2 + 7 z 2 + a 5 z + a z + 2 z a -1 + a 4 - a 5 z -1 - a 3 z -1$ (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 =

-3 is the signature of L11n87. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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