L11n128

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_22,10,7,9 X₂₇₃₈ X_15,5,16,4 X_5,13,6,12 X_16,12,17,11 X_6,18,1,17 X_19,14,20,15 X_13,20,14,21 X_21,19,22,18
Gauss code	{1, -4, 2, 5, -6, -8}, {4, -1, 3, -2, 7, 6, -10, 9, -5, -7, 8, 11, -9, 10, -11, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 + 3 v 2 u 3 -2 v u 3 -2 v 2 u 2 + v u 2 -2 u 2 -2 v u + 3 u + v -1$ (db)
Jones polynomial	$q^{13/2}-3 q^{11/2}+4 q^{9/2}-6 q^{7/2}+6 q^{5/2}-6 q^{3/2}+5 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 -2 z 5 a -1 + 5 z 5 a -3 - z 5 a -5 + a z 3 -8 z 3 a -1 + 8 z 3 a -3 -4 z 3 a -5 + 3 a z -7 z a -1 + 7 z a -3 -3 z a -5 + z a -7 + a z -1 -2 a -1 z -1 + 2 a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$- z 9 a -1 - z 9 a -3 -5 z 8 a -2 -3 z 8 a -4 -2 z 8 - a z 7 -2 z 7 a -3 -3 z 7 a -5 + 20 z 6 a -2 + 10 z 6 a -4 - z 6 a -6 + 9 z 6 + 5 a z 5 + 15 z 5 a -1 + 20 z 5 a -3 + 10 z 5 a -5 -20 z 4 a -2 -9 z 4 a -4 -11 z 4 -8 a z 3 -25 z 3 a -1 -27 z 3 a -3 -13 z 3 a -5 -3 z 3 a -7 + 6 z 2 a -2 + 3 z 2 a -4 - z 2 a -8 + 4 z 2 + 5 a z + 13 z a -1 + 13 z a -3 + 7 z a -5 + 2 z a -7 - a -2 - a z -1 -2 a -1 z -1 -2 a -3 z -1 - a -5 z -1$ (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 =

3 is the signature of L11n128. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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