L11n136

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_11,19,12,18 X_3,10,4,11 X_2,17,3,18 X_12,5,13,6 X₆₇₁₈ X_16,10,17,9 X_20,14,21,13 X_22,16,7,15 X_19,4,20,5 X_14,22,15,21
Gauss code	{1, -4, -3, 10, 5, -6}, {6, -1, 7, 3, -2, -5, 8, -11, 9, -7, 4, 2, -10, -8, 11, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 6 + v u 5 - u 5 - v u 4 + v u 3 - v u 2 - v 2 u + v u - v$ (db)
Jones polynomial	$q^{7/2}-q^{5/2}+2 q^{3/2}-3 \sqrt{q}+\frac{2}{\sqrt{q}}-\frac{3}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{2}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- a z 7 + a 3 z 5 -7 a z 5 + z 5 a -1 + 5 a 3 z 3 -17 a z 3 + 5 z 3 a -1 + 7 a 3 z -17 a z + 7 z a -1 + 3 a 3 z -1 -5 a z -1 + 2 a -1 z -1$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 - a 4 z 8 -2 a 2 z 8 - z 8 - a 5 z 7 + 6 a 3 z 7 + 7 a z 7 + 5 a 4 z 6 + 11 a 2 z 6 + 6 z 6 + 6 a 5 z 5 -13 a 3 z 5 -21 a z 5 -2 z 5 a -1 -6 a 4 z 4 -20 a 2 z 4 - z 4 a -2 -15 z 4 -10 a 5 z 3 + 16 a 3 z 3 + 33 a z 3 + 6 z 3 a -1 - z 3 a -3 + a 4 z 2 + 17 a 2 z 2 - z 2 a -4 + 17 z 2 + 4 a 5 z -11 a 3 z -22 a z -7 z a -1 -5 a 2 + a -4 -5 + 3 a 3 z -1 + 5 a z -1 + 2 a -1 z -1$ (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 L11n136. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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