L11n191

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_22,10,7,9 X_10,14,11,13 X_16,6,17,5 X_15,21,16,20 X_21,19,22,18 X_19,15,20,14 X₂₇₃₈ X_4,11,5,12 X_6,18,1,17
Gauss code	{1, -9, 2, -10, 5, -11}, {9, -1, 3, -4, 10, -2, 4, 8, -6, -5, 11, 7, -8, 6, -7, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 2 v u 4 - u 4 + 2 v 2 u 3 -2 v u 3 + u 3 - v 2 u 2 + v u 2 - u 2 + v 2 u -2 v u + 2 u - v 2 + 2 v -1$ (db)
Jones polynomial	$q^{13/2}-3 q^{11/2}+5 q^{9/2}-6 q^{7/2}+7 q^{5/2}-7 q^{3/2}+5 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 -2 z 5 a -1 + 5 z 5 a -3 - z 5 a -5 + a z 3 -8 z 3 a -1 + 8 z 3 a -3 -4 z 3 a -5 + 3 a z -8 z a -1 + 6 z a -3 -3 z a -5 + z a -7 + 2 a z -1 -3 a -1 z -1 + a -3 z -1$ (db)
Kauffman polynomial	$- z 9 a -1 - z 9 a -3 -5 z 8 a -2 -3 z 8 a -4 -2 z 8 - a z 7 - z 7 a -1 -3 z 7 a -3 -3 z 7 a -5 + 17 z 6 a -2 + 8 z 6 a -4 - z 6 a -6 + 8 z 6 + 5 a z 5 + 18 z 5 a -1 + 21 z 5 a -3 + 8 z 5 a -5 -10 z 4 a -2 -4 z 4 a -4 - z 4 a -6 -7 z 4 -9 a z 3 -27 z 3 a -1 -23 z 3 a -3 -8 z 3 a -5 -3 z 3 a -7 -4 z 2 a -2 + z 2 a -6 - z 2 a -8 -2 z 2 + 7 a z + 13 z a -1 + 8 z a -3 + 3 z a -5 + z a -7 + 3 a -2 + a -4 + 3-2 a z -1 -3 a -1 z -1 - a -3 z -1$ (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 L11n191. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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