L11a223

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 2 v u 4 - u 4 + 4 v 2 u 3 -10 v u 3 + 5 u 3 -7 v 2 u 2 + 17 v u 2 -7 u 2 + 5 v 2 u -10 v u + 4 u - v 2 + 2 v -1$ (db)
Jones polynomial	$q^{15/2}-4 q^{13/2}+9 q^{11/2}-16 q^{9/2}+21 q^{7/2}-25 q^{5/2}+25 q^{3/2}-22 \sqrt{q}+\frac{16}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + a z 5 -4 z 5 a -1 + 3 z 5 a -3 + 2 a z 3 -9 z 3 a -1 + 7 z 3 a -3 -3 z 3 a -5 + 3 a z -7 z a -1 + 7 z a -3 -3 z a -5 + z a -7 + a z -1 -2 a -1 z -1 + 2 a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -2 -2 z 10 a -4 -7 z 9 a -1 -13 z 9 a -3 -6 z 9 a -5 -20 z 8 a -2 -16 z 8 a -4 -7 z 8 a -6 -11 z 8 -9 a z 7 -6 z 7 a -1 + 9 z 7 a -3 + 2 z 7 a -5 -4 z 7 a -7 -4 a 2 z 6 + 45 z 6 a -2 + 41 z 6 a -4 + 14 z 6 a -6 - z 6 a -8 + 15 z 6 - a 3 z 5 + 14 a z 5 + 33 z 5 a -1 + 27 z 5 a -3 + 18 z 5 a -5 + 9 z 5 a -7 + 4 a 2 z 4 -28 z 4 a -2 -26 z 4 a -4 -8 z 4 a -6 + 2 z 4 a -8 -8 z 4 + a 3 z 3 -11 a z 3 -32 z 3 a -1 -32 z 3 a -3 -19 z 3 a -5 -7 z 3 a -7 - a 2 z 2 + 6 z 2 a -2 + 5 z 2 a -4 + z 2 a -6 - z 2 a -8 + 2 z 2 + 5 a z + 13 z a -1 + 13 z a -3 + 7 z a -5 + 2 z a -7 - a -2 - a z -1 -2 a -1 z -1 -2 a -3 z -1 - a -5 z -1$ (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 L11a223. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -4$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 0$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{10}$
$r = 1$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 2$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{14}$
$r = 3$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 4$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 6$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$