L11n225

(Knotscape image)

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

Planar diagram presentation	X_10,1,11,2 X_3,12,4,13 X_16,9,17,10 X_11,20,12,21 X_22,15,9,16 X_5,14,6,15 X_13,4,14,5 X_6,20,7,19 X_18,8,19,7 X_8,18,1,17 X_21,3,22,2
Gauss code	{1, 11, -2, 7, -6, -8, 9, -10}, {3, -1, -4, 2, -7, 6, 5, -3, 10, -9, 8, 4, -11, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 + v u 3 -2 v 3 u 2 + 4 v 2 u 2 -4 v u 2 + 2 u 2 + 2 v 3 u -4 v 2 u + 4 v u -2 u + v 2 - v$ (db)
Jones polynomial	$-2 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{10}{q^{7/2}}+\frac{9}{q^{9/2}}-\frac{8}{q^{11/2}}+\frac{6}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z 3 a 7 - z a 7 + z 5 a 5 + z 3 a 5 -2 z a 5 - a 5 z -1 + 2 z 5 a 3 + 7 z 3 a 3 + 8 z a 3 + 3 a 3 z -1 -2 z 3 a -5 z a -2 a z -1$ (db)
Kauffman polynomial	$- z 4 a 10 + z 2 a 10 -3 z 5 a 9 + 3 z 3 a 9 -5 z 6 a 8 + 7 z 4 a 8 -3 z 2 a 8 -5 z 7 a 7 + 8 z 5 a 7 -7 z 3 a 7 + 2 z a 7 -3 z 8 a 6 + 3 z 6 a 6 -4 z 4 a 6 + 4 z 2 a 6 - a 6 - z 9 a 5 -2 z 7 a 5 + 2 z 5 a 5 + 4 z 3 a 5 -4 z a 5 + a 5 z -1 -4 z 8 a 4 + 10 z 6 a 4 -14 z 4 a 4 + 12 z 2 a 4 -3 a 4 - z 9 a 3 + 3 z 7 a 3 -12 z 5 a 3 + 24 z 3 a 3 -15 z a 3 + 3 a 3 z -1 - z 8 a 2 + 2 z 6 a 2 -2 z 4 a 2 + 4 z 2 a 2 -3 a 2 -3 z 5 a + 10 z 3 a -9 z a + 2 a z -1$ (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 L11n225. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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