L11n15

(Knotscape image)

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

Visit L11n15's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u + u + v -1$ (db)
Jones polynomial	$-\frac{1}{\sqrt{q}}-\frac{1}{q^{5/2}}-\frac{1}{q^{9/2}}+\frac{1}{q^{11/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{17/2}}+\frac{1}{q^{19/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 9 - a 9 z -1 + z 3 a 7 + 3 z a 7 + 2 a 7 z -1 - z a 5 - a 5 z -1 + a 3 z -1 - z a - a z -1$ (db)
Kauffman polynomial	$- z 8 a 10 + 7 z 6 a 10 -15 z 4 a 10 + 11 z 2 a 10 -3 a 10 - z 9 a 9 + 7 z 7 a 9 -15 z 5 a 9 + 11 z 3 a 9 -3 z a 9 + a 9 z -1 -2 z 8 a 8 + 14 z 6 a 8 -31 z 4 a 8 + 26 z 2 a 8 -7 a 8 - z 9 a 7 + 7 z 7 a 7 -16 z 5 a 7 + 16 z 3 a 7 -8 z a 7 + 2 a 7 z -1 - z 8 a 6 + 7 z 6 a 6 -16 z 4 a 6 + 15 z 2 a 6 -4 a 6 - z 5 a 5 + 5 z 3 a 5 -5 z a 5 + a 5 z -1 - z a 3 + a 3 z -1 - a 2 - z a + a z -1$ (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 =

-1 is the signature of L11n15. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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