L11a225

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_16,6,17,5 X_18,10,19,9 X_10,20,11,19 X_2,11,3,12 X_12,3,13,4 X₄₇₅₈ X_20,16,21,15 X_22,14,7,13 X_14,22,15,21 X_6,18,1,17
Gauss code	{1, -5, 6, -7, 2, -11}, {7, -1, 3, -4, 5, -6, 9, -10, 8, -2, 11, -3, 4, -8, 10, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 4 + 2 v u 4 + 4 v 2 u 3 -8 v u 3 + 2 u 3 -4 v 2 u 2 + 11 v u 2 -4 u 2 + 2 v 2 u -8 v u + 4 u + 2 v -2$ (db)
Jones polynomial	$q^{15/2}-3 q^{13/2}+7 q^{11/2}-11 q^{9/2}+15 q^{7/2}-18 q^{5/2}+17 q^{3/2}-16 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 - z 7 a -3 + a z 5 -4 z 5 a -1 -4 z 5 a -3 + z 5 a -5 + 3 a z 3 -6 z 3 a -1 -6 z 3 a -3 + 3 z 3 a -5 + 3 a z -3 z a -1 -4 z a -3 + 3 z a -5 + a z -1 -2 a -3 z -1 + a -5 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -2 -2 z 10 a -4 -5 z 9 a -1 -10 z 9 a -3 -5 z 9 a -5 -5 z 8 a -2 -4 z 8 a -4 -5 z 8 a -6 -6 z 8 -5 a z 7 + 7 z 7 a -1 + 28 z 7 a -3 + 13 z 7 a -5 -3 z 7 a -7 -3 a 2 z 6 + 22 z 6 a -2 + 24 z 6 a -4 + 14 z 6 a -6 - z 6 a -8 + 10 z 6 - a 3 z 5 + 8 a z 5 - z 5 a -1 -32 z 5 a -3 -14 z 5 a -5 + 8 z 5 a -7 + 5 a 2 z 4 -30 z 4 a -2 -35 z 4 a -4 -13 z 4 a -6 + 3 z 4 a -8 -6 z 4 + 2 a 3 z 3 -4 a z 3 -8 z 3 a -1 + 14 z 3 a -3 + 12 z 3 a -5 -4 z 3 a -7 - a 2 z 2 + 13 z 2 a -2 + 22 z 2 a -4 + 9 z 2 a -6 -2 z 2 a -8 + z 2 - a 3 z + 4 a z + 4 z a -1 -5 z a -3 -4 z a -5 -3 a -2 -5 a -4 -2 a -6 + 1- a z -1 + 2 a -3 z -1 + a -5 z -1$ (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 =

1 is the signature of L11a225. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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