L11n91

(Knotscape image)

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

Visit L11n91's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_7,16,8,17 X_11,20,12,21 X_17,22,18,5 X_21,18,22,19 X_19,10,20,11 X_9,14,10,15 X_15,8,16,9 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, -8, 7, -4, -2, 11, 8, -9, 3, -5, 6, -7, 4, -6, 5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + u 3 + v u 2 - u 2 - v u + u + v -1$ (db)
Jones polynomial	$-\sqrt{q}+\frac{2}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{2}{q^{11/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{17/2}}+\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z a 9 - a 9 z -1 + z 3 a 7 + 3 z a 7 + 2 a 7 z -1 - z 3 a 5 -2 z a 5 - a 5 z -1 + z 5 a 3 + 3 z 3 a 3 + 2 z a 3 + a 3 z -1 - z 3 a -2 z a - a z -1$ (db)
Kauffman polynomial	$- z 8 a 10 + 7 z 6 a 10 -15 z 4 a 10 + 12 z 2 a 10 -3 a 10 - z 9 a 9 + 7 z 7 a 9 -14 z 5 a 9 + 10 z 3 a 9 -3 z a 9 + a 9 z -1 -2 z 8 a 8 + 16 z 6 a 8 -36 z 4 a 8 + 27 z 2 a 8 -7 a 8 - z 9 a 7 + 8 z 7 a 7 -18 z 5 a 7 + 14 z 3 a 7 -8 z a 7 + 2 a 7 z -1 - z 8 a 6 + 8 z 6 a 6 -20 z 4 a 6 + 15 z 2 a 6 -4 a 6 -4 z 5 a 5 + 10 z 3 a 5 -7 z a 5 + a 5 z -1 -3 z 6 a 4 + 6 z 4 a 4 - z 2 a 4 - z 7 a 3 - z 5 a 3 + 9 z 3 a 3 -5 z a 3 + a 3 z -1 -2 z 6 a 2 + 5 z 4 a 2 - z 2 a 2 - a 2 - z 5 a + 3 z 3 a -3 z a + a z -1$ (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 L11n91. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$	$i = 0$
$r = -9$		${\mathbb Z}$
$r = -8$		${\mathbb Z}_2$	${\mathbb Z}$
$r = -7$		${\mathbb Z}$
$r = -6$		${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{2}$	${\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -1$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 0$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}^{3}$
$r = 1$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 2$	${\mathbb Z}_2$	${\mathbb Z}$