L11n40

(Knotscape image)

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

Visit L11n40's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_9,18,10,19 X₃₈₄₉ X_5,14,6,15 X_15,20,16,21 X_17,22,18,5 X_21,16,22,17 X_19,10,20,11 X_11,2,12,3
Gauss code	{1, 11, -5, -3}, {-6, -1, 2, 5, -4, 10, -11, -2, 3, 6, -7, 9, -8, 4, -10, 7, -9, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 5 + 3 v u 4 - u 4 -3 v u 3 + 2 u 3 + 2 v u 2 -3 u 2 - v u + 3 u -2$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{4}{q^{11/2}}+\frac{5}{q^{13/2}}-\frac{7}{q^{15/2}}+\frac{7}{q^{17/2}}-\frac{7}{q^{19/2}}+\frac{6}{q^{21/2}}-\frac{4}{q^{23/2}}+\frac{2}{q^{25/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$- z a 13 - a 13 z -1 + z 5 a 11 + 4 z 3 a 11 + 4 z a 11 + a 11 z -1 - z 7 a 9 -4 z 5 a 9 -2 z 3 a 9 + 4 z a 9 + 2 a 9 z -1 - z 7 a 7 -6 z 5 a 7 -12 z 3 a 7 -9 z a 7 -2 a 7 z -1$ (db)
Kauffman polynomial	$-3 z 2 a 16 + a 16 - z 5 a 15 -3 z 3 a 15 + z a 15 -3 z 6 a 14 + 3 z 4 a 14 -2 z 2 a 14 -4 z 7 a 13 + 8 z 5 a 13 -4 z 3 a 13 - z a 13 + a 13 z -1 -3 z 8 a 12 + 7 z 6 a 12 -7 z 4 a 12 + 9 z 2 a 12 -3 a 12 - z 9 a 11 - z 7 a 11 + 7 z 5 a 11 + z 3 a 11 -4 z a 11 + a 11 z -1 -4 z 8 a 10 + 13 z 6 a 10 -9 z 4 a 10 + z 2 a 10 - z 9 a 9 + 2 z 7 a 9 + 4 z 5 a 9 -10 z 3 a 9 + 7 z a 9 -2 a 9 z -1 - z 8 a 8 + 3 z 6 a 8 + z 4 a 8 -7 z 2 a 8 + 3 a 8 - z 7 a 7 + 6 z 5 a 7 -12 z 3 a 7 + 9 z a 7 -2 a 7 z -1$ (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 =

-7 is the signature of L11n40. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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