L11n24

(Knotscape image)

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

Visit L11n24's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + u 3 + 2 v u 2 -2 u 2 -2 v u + 2 u + v -1$ (db)
Jones polynomial	$-q^{11/2}+q^{9/2}-q^{5/2}+3 q^{3/2}-5 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a z 5 - a 3 z 3 + 3 a z 3 -2 z 3 a -1 + z 3 a -3 - a 3 z + 3 a z -5 z a -1 + 4 z a -3 - z a -5 + 2 a z -1 -4 a -1 z -1 + 3 a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$- a z 9 - z 9 a -1 -3 a 2 z 8 -2 z 8 a -2 - z 8 a -4 -4 z 8 -3 a 3 z 7 + 3 z 7 a -1 - z 7 a -3 - z 7 a -5 - a 4 z 6 + 11 a 2 z 6 + 15 z 6 a -2 + 7 z 6 a -4 + 20 z 6 + 11 a 3 z 5 + 15 a z 5 + 8 z 5 a -1 + 10 z 5 a -3 + 6 z 5 a -5 + 3 a 4 z 4 -7 a 2 z 4 -31 z 4 a -2 -13 z 4 a -4 -28 z 4 -8 a 3 z 3 -23 a z 3 -27 z 3 a -1 -21 z 3 a -3 -9 z 3 a -5 - a 4 z 2 + a 2 z 2 + 20 z 2 a -2 + 9 z 2 a -4 + 13 z 2 + 3 a 3 z + 11 a z + 18 z a -1 + 14 z a -3 + 4 z a -5 - a 2 -3 a -2 - a -4 -2-2 a z -1 -4 a -1 z -1 -3 a -3 z -1 - a -5 z -1$ (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 L11n24. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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