L11n230

(Knotscape image)

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

Planar diagram presentation	X_10,1,11,2 X_11,17,12,16 X_5,21,6,20 X_3,12,4,13 X_7,14,8,15 X_13,6,14,7 X_17,9,18,22 X_21,19,22,18 X_8,9,1,10 X_19,5,20,4 X_15,2,16,3
Gauss code	{1, 11, -4, 10, -3, 6, -5, -9}, {9, -1, -2, 4, -6, 5, -11, 2, -7, 8, -10, 3, -8, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 + v u 3 -2 v 3 u 2 + 4 v 2 u 2 -5 v u 2 + 2 u 2 + 2 v 3 u -5 v 2 u + 4 v u -2 u + v 2 - v$ (db)
Jones polynomial	$2 q^{5/2}-5 q^{3/2}+7 \sqrt{q}-\frac{10}{\sqrt{q}}+\frac{10}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 3 a 5 + 2 z a 5 + a 5 z -1 - z 5 a 3 -2 z 3 a 3 -2 z a 3 - a 3 z -1 -2 z 5 a -6 z 3 a -5 z a + 2 z 3 a -1 + 3 z a -1$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 -3 a 4 z 8 -5 a 2 z 8 -2 z 8 -3 a 5 z 7 -4 a 3 z 7 -2 a z 7 - z 7 a -1 -2 a 6 z 6 + 6 a 4 z 6 + 10 a 2 z 6 + 2 z 6 - a 7 z 5 + 6 a 5 z 5 + 13 a 3 z 5 + 3 a z 5 -3 z 5 a -1 + 4 a 6 z 4 -7 a 4 z 4 -11 a 2 z 4 -3 z 4 a -2 -3 z 4 + 3 a 7 z 3 -5 a 5 z 3 -16 a 3 z 3 + 8 z 3 a -1 - a 6 z 2 + 3 a 4 z 2 + 4 a 2 z 2 + 4 z 2 a -2 + 4 z 2 -2 a 7 z + 4 a 5 z + 7 a 3 z -2 a z -3 z a -1 + a 4 - a 5 z -1 - a 3 z -1$ (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 L11n230. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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