L11n232

(Knotscape image)

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

Planar diagram presentation	X_10,1,11,2 X_18,11,19,12 X_8,9,1,10 X_22,19,9,20 X_20,6,21,5 X_4,22,5,21 X_7,15,8,14 X_12,4,13,3 X_13,16,14,17 X_15,7,16,6 X_2,18,3,17
Gauss code	{1, -11, 8, -6, 5, 10, -7, -3}, {3, -1, 2, -8, -9, 7, -10, 9, 11, -2, 4, -5, 6, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 3 -2 v u 3 + u 3 - v 2 u 2 + v u 2 - u 2 - v 3 u + v 2 u - v u + v 3 -2 v 2 + v$ (db)
Jones polynomial	$-q^{11/2}+2 q^{9/2}-4 q^{7/2}+4 q^{5/2}-5 q^{3/2}+4 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z 5 a -1 + a z 3 -3 z 3 a -1 + 2 z 3 a -3 + a z -2 z a -1 + 4 z a -3 - z a -5 + a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$- z 9 a -1 - z 9 a -3 -4 z 8 a -2 -2 z 8 a -4 -2 z 8 - a z 7 + z 7 a -1 + z 7 a -3 - z 7 a -5 + 17 z 6 a -2 + 9 z 6 a -4 + 8 z 6 + 4 a z 5 + 10 z 5 a -1 + 11 z 5 a -3 + 5 z 5 a -5 -17 z 4 a -2 -10 z 4 a -4 -7 z 4 -5 a z 3 -15 z 3 a -1 -18 z 3 a -3 -8 z 3 a -5 - a 2 z 2 + 4 z 2 a -2 + 2 z 2 a -4 + z 2 + 2 a z + 5 z a -1 + 8 z a -3 + 5 z a -5 + a -4 - a -3 z -1 - a -5 z -1$ (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 L11n232. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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