L11n215

(Knotscape image)

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

Planar diagram presentation	X_10,1,11,2 X_12,3,13,4 X_14,5,15,6 X_9,18,10,19 X_17,22,18,9 X_21,1,22,8 X_20,15,21,16 X_7,16,8,17 X_6,20,7,19 X_4,11,5,12 X_2,13,3,14
Gauss code	{1, -11, 2, -10, 3, -9, -8, 6}, {-4, -1, 10, -2, 11, -3, 7, 8, -5, 4, 9, -7, -6, 5}

A Braid Representative

A Morse Link Presentation

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

-3 is the signature of L11n215. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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