L11n6

(Knotscape image)

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

Visit L11n6's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_9,14,10,15 X₈₄₉₃ X_5,11,6,10 X_11,19,12,18 X_13,20,14,21 X_19,5,20,22 X_21,12,22,13 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {-6, -1, 2, -5, -4, 6, -7, 10, -8, 4, 11, -2, 3, 7, -9, 8, -10, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + u 3 + 4 v u 2 -4 u 2 -4 v u + 4 u + v -1$ (db)
Jones polynomial	$q^{9/2}-3 q^{7/2}+4 q^{5/2}-6 q^{3/2}+6 \sqrt{q}-\frac{7}{\sqrt{q}}+\frac{6}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{1}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 5 z -1 -3 z a 3 -3 a 3 z -1 + 3 z 3 a + 6 z a + 4 a z -1 - z 5 a -1 -3 z 3 a -1 -4 z a -1 -2 a -1 z -1 + z 3 a -3 + z a -3$ (db)
Kauffman polynomial	$- a z 9 - z 9 a -1 -2 a 2 z 8 -3 z 8 a -2 -5 z 8 - a 3 z 7 -2 z 7 a -1 -3 z 7 a -3 + 9 a 2 z 6 + 10 z 6 a -2 - z 6 a -4 + 20 z 6 + 4 a 3 z 5 + 13 a z 5 + 20 z 5 a -1 + 11 z 5 a -3 -2 a 4 z 4 -18 a 2 z 4 -6 z 4 a -2 + 3 z 4 a -4 -25 z 4 - a 5 z 3 -12 a 3 z 3 -30 a z 3 -28 z 3 a -1 -9 z 3 a -3 + 3 a 4 z 2 + 12 a 2 z 2 + z 2 a -2 - z 2 a -4 + 11 z 2 + 2 a 5 z + 12 a 3 z + 20 a z + 13 z a -1 + 3 z a -3 - a 4 -3 a 2 - a -2 -2- a 5 z -1 -3 a 3 z -1 -4 a z -1 -2 a -1 z -1$ (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 L11n6. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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