L11n213

(Knotscape image)

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

Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 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_4,13,5,14 X_6,20,7,19
Gauss code	{1, -2, 3, -10, 4, -11, -9, 7}, {-5, -1, 2, -3, 10, -4, 8, 9, -6, 5, 11, -8, -7, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + 2 v 2 u 3 + v 3 u 2 -3 v 2 u 2 + 2 v u 2 + 2 v 2 u -3 v u + u + 2 v -1$ (db)
Jones polynomial	$-\frac{2}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{6}{q^{13/2}}+\frac{6}{q^{15/2}}-\frac{5}{q^{17/2}}+\frac{3}{q^{19/2}}-\frac{2}{q^{21/2}}+\frac{1}{q^{23/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- z 5 a 9 -4 z 3 a 9 -4 z a 9 - a 9 z -1 + z 7 a 7 + 6 z 5 a 7 + 13 z 3 a 7 + 12 z a 7 + 3 a 7 z -1 -2 z 5 a 5 -9 z 3 a 5 -10 z a 5 -2 a 5 z -1$ (db)
Kauffman polynomial	$- z 4 a 14 + 2 z 2 a 14 -2 z 5 a 13 + 4 z 3 a 13 - z a 13 -2 z 6 a 12 + 3 z 4 a 12 - z 2 a 12 -2 z 7 a 11 + 4 z 5 a 11 -4 z 3 a 11 -2 z 8 a 10 + 7 z 6 a 10 -12 z 4 a 10 + 5 z 2 a 10 - a 10 - z 9 a 9 + 3 z 7 a 9 -5 z 5 a 9 + 3 z 3 a 9 -3 z a 9 + a 9 z -1 -3 z 8 a 8 + 13 z 6 a 8 -23 z 4 a 8 + 17 z 2 a 8 -3 a 8 - z 9 a 7 + 5 z 7 a 7 -14 z 5 a 7 + 23 z 3 a 7 -15 z a 7 + 3 a 7 z -1 - z 8 a 6 + 4 z 6 a 6 -7 z 4 a 6 + 9 z 2 a 6 -3 a 6 -3 z 5 a 5 + 12 z 3 a 5 -11 z a 5 + 2 a 5 z -1$ (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 =

-5 is the signature of L11n213. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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