L11n89

(Knotscape image)

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

Visit L11n89's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_7,16,8,17 X_17,22,18,5 X_9,15,10,14 X_19,10,20,11 X_21,9,22,8 X_13,18,14,19 X_15,21,16,20 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 7, -5, 6, 11, -2, -8, 5, -9, 3, -4, 8, -6, 9, -7, 4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 3 v u 4 + u 4 -2 v u 3 -2 u 3 -2 v u 2 -2 u 2 + v u + 3 u -1$ (db)
Jones polynomial	$q^{3/2}-3 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{2}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{2}{q^{13/2}}+\frac{2}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a 9 z -1 -4 z a 7 -3 a 7 z -1 + z 5 a 5 + 6 z 3 a 5 + 8 z a 5 + 4 a 5 z -1 - z 7 a 3 -5 z 5 a 3 -7 z 3 a 3 -6 z a 3 -2 a 3 z -1 + z 5 a + 3 z 3 a$ (db)
Kauffman polynomial	$- z 7 a 9 + 5 z 5 a 9 -6 z 3 a 9 + 4 z a 9 - a 9 z -1 -2 z 8 a 8 + 11 z 6 a 8 -14 z 4 a 8 + 6 z 2 a 8 - a 8 - z 9 a 7 + 4 z 7 a 7 + 6 z 5 a 7 -24 z 3 a 7 + 16 z a 7 -3 a 7 z -1 -3 z 8 a 6 + 21 z 6 a 6 -36 z 4 a 6 + 18 z 2 a 6 -3 a 6 - z 9 a 5 + 3 z 7 a 5 + 14 z 5 a 5 -40 z 3 a 5 + 25 z a 5 -4 a 5 z -1 -3 z 8 a 4 + 18 z 6 a 4 -28 z 4 a 4 + 15 z 2 a 4 -2 a 4 -5 z 7 a 3 + 25 z 5 a 3 -32 z 3 a 3 + 15 z a 3 -2 a 3 z -1 -2 z 8 a 2 + 7 z 6 a 2 -3 z 4 a 2 + 2 z 2 a 2 - a 2 -3 z 7 a + 12 z 5 a -10 z 3 a + 2 z a - z 6 + 3 z 4 - z 2$ (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 L11n89. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$	$i = 0$
$r = -8$		${\mathbb Z}$
$r = -7$		${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$		${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$		${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -3$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 0$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{4}$
$r = 1$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$