L11n223

(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_18,8,19,7 X_16,9,17,10 X_22,15,9,16 X_21,5,22,4 X_5,14,6,15 X_13,20,14,21 X_8,18,1,17 X_6,20,7,19
Gauss code	{1, -2, 3, 7, -8, -11, 4, -10}, {5, -1, 2, -3, -9, 8, 6, -5, 10, -4, 11, 9, -7, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 + v u 3 -2 v 3 u 2 + 3 v 2 u 2 -3 v u 2 + 2 u 2 + 2 v 3 u -3 v 2 u + 3 v u -2 u + v 2 - v$ (db)
Jones polynomial	$-2 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{8}{q^{9/2}}-\frac{7}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z 3 a 7 -2 z a 7 + z 5 a 5 + 2 z 3 a 5 - a 5 z -1 + 2 z 5 a 3 + 7 z 3 a 3 + 7 z a 3 + 3 a 3 z -1 -2 z 3 a -5 z a -2 a z -1$ (db)
Kauffman polynomial	$- z 4 a 10 + 2 z 2 a 10 -2 z 5 a 9 + 3 z 3 a 9 -3 z 6 a 8 + 5 z 4 a 8 -3 z 2 a 8 -4 z 7 a 7 + 11 z 5 a 7 -15 z 3 a 7 + 4 z a 7 -3 z 8 a 6 + 8 z 6 a 6 -12 z 4 a 6 + 5 z 2 a 6 - a 6 - z 9 a 5 - z 7 a 5 + 6 z 5 a 5 -7 z 3 a 5 + a 5 z -1 -4 z 8 a 4 + 13 z 6 a 4 -20 z 4 a 4 + 16 z 2 a 4 -3 a 4 - z 9 a 3 + 3 z 7 a 3 -10 z 5 a 3 + 21 z 3 a 3 -13 z a 3 + 3 a 3 z -1 - z 8 a 2 + 2 z 6 a 2 -2 z 4 a 2 + 6 z 2 a 2 -3 a 2 -3 z 5 a + 10 z 3 a -9 z a + 2 a z -1$ (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 =

-3 is the signature of L11n223. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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