L11n19

(Knotscape image)

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

Visit L11n19's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_5,12,6,13 X₃₈₄₉ X_13,22,14,5 X_21,14,22,15 X_9,18,10,19 X_11,20,12,21 X_19,10,20,11 X_15,2,16,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -8, 10, -9, 4, -6, 7, -11, -2, 3, 8, -10, 9, -7, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 5 + v u 4 + u -2$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{2}{q^{11/2}}+\frac{1}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{1}{q^{17/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{21/2}}+\frac{1}{q^{25/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$- z a 13 -2 a 13 z -1 + z 5 a 11 + 6 z 3 a 11 + 9 z a 11 + 4 a 11 z -1 - z 7 a 9 -6 z 5 a 9 -10 z 3 a 9 -5 z a 9 - a 9 z -1 - z 7 a 7 -6 z 5 a 7 -10 z 3 a 7 -5 z a 7 - a 7 z -1$ (db)
Kauffman polynomial	$- z 2 a 16 + 2 a 16 - z a 15 - z 2 a 14 + a 14 - z 5 a 13 + 5 z 3 a 13 -7 z a 13 + 2 a 13 z -1 - z 8 a 12 + 7 z 6 a 12 -17 z 4 a 12 + 18 z 2 a 12 -6 a 12 - z 9 a 11 + 7 z 7 a 11 -18 z 5 a 11 + 24 z 3 a 11 -15 z a 11 + 4 a 11 z -1 -2 z 8 a 10 + 12 z 6 a 10 -22 z 4 a 10 + 17 z 2 a 10 -5 a 10 - z 9 a 9 + 6 z 7 a 9 -11 z 5 a 9 + 9 z 3 a 9 -4 z a 9 + a 9 z -1 - z 8 a 8 + 5 z 6 a 8 -5 z 4 a 8 - z 2 a 8 + a 8 - z 7 a 7 + 6 z 5 a 7 -10 z 3 a 7 + 5 z a 7 - a 7 z -1$ (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 L11n19. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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