L11n231

(Knotscape image)

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

Planar diagram presentation	X_10,1,11,2 X_16,11,17,12 X_5,21,6,20 X_12,4,13,3 X_14,8,15,7 X_6,14,7,13 X_17,9,18,22 X_21,19,22,18 X_8,9,1,10 X_19,5,20,4 X_2,16,3,15
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 $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + 2 v 2 u 3 - v u 3 + v 3 u 2 -5 v 2 u 2 + 3 v u 2 + 3 v 2 u -5 v u + u - v 2 + 2 v -1$ (db)
Jones polynomial	$-q^{15/2}+3 q^{13/2}-6 q^{11/2}+8 q^{9/2}-9 q^{7/2}+8 q^{5/2}-8 q^{3/2}+5 \sqrt{q}-\frac{3}{\sqrt{q}}+\frac{1}{q^{3/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 - z 5 a -1 + 5 z 5 a -3 - z 5 a -5 -3 z 3 a -1 + 9 z 3 a -3 -3 z 3 a -5 -2 z a -1 + 7 z a -3 -3 z a -5 + a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$- z 9 a -3 - z 9 a -5 -3 z 8 a -2 -5 z 8 a -4 -2 z 8 a -6 -3 z 7 a -1 -4 z 7 a -3 -2 z 7 a -5 - z 7 a -7 + 7 z 6 a -2 + 12 z 6 a -4 + 4 z 6 a -6 - z 6 + 10 z 5 a -1 + 20 z 5 a -3 + 10 z 5 a -5 -6 z 4 a -4 -6 z 4 a -6 -3 z 4 a -8 + 3 z 4 -9 z 3 a -1 -20 z 3 a -3 -13 z 3 a -5 -3 z 3 a -7 - z 3 a -9 -2 z 2 a -2 - z 2 a -4 + z 2 a -6 + 2 z 2 a -8 -2 z 2 + 3 z a -1 + 9 z a -3 + 6 z a -5 + z a -7 + z a -9 + a -4 - a -3 z -1 - a -5 z -1$ (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 =

3 is the signature of L11n231. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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