L11n27

(Knotscape image)

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

Visit L11n27's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_17,1,18,4 X_5,12,6,13 X₃₈₄₉ X_13,22,14,5 X_21,14,22,15 X_11,18,12,19 X_9,20,10,21 X_19,10,20,11 X_2,16,3,15
Gauss code	{1, -11, -5, 3}, {-4, -1, 2, 5, -9, 10, -8, 4, -6, 7, 11, -2, -3, 8, -10, 9, -7, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-3 v u + 3 u + 3 v -3$ (db)
Jones polynomial	$-\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{1}{q^{17/2}}+\frac{1}{q^{19/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 9 - a 9 z -1 + z 3 a 7 + z a 7 + a 7 z -1 + 2 z 3 a 5 + 4 z a 5 + 2 a 5 z -1 -3 z a 3 -2 a 3 z -1 - z a$ (db)
Kauffman polynomial	$- z 8 a 10 + 7 z 6 a 10 -17 z 4 a 10 + 16 z 2 a 10 -4 a 10 - z 9 a 9 + 5 z 7 a 9 -6 z 5 a 9 + a 9 z -1 -4 z 8 a 8 + 23 z 6 a 8 -42 z 4 a 8 + 31 z 2 a 8 -9 a 8 - z 9 a 7 + z 7 a 7 + 13 z 5 a 7 -22 z 3 a 7 + 5 z a 7 + a 7 z -1 -3 z 8 a 6 + 14 z 6 a 6 -18 z 4 a 6 + 11 z 2 a 6 -4 a 6 -4 z 7 a 5 + 19 z 5 a 5 -25 z 3 a 5 + 13 z a 5 -2 a 5 z -1 -2 z 6 a 4 + 7 z 4 a 4 -5 z 2 a 4 + 2 a 4 -3 z 3 a 3 + 7 z a 3 -2 a 3 z -1 - z 2 a 2 - z a$ (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 L11n27. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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