L11n138

(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_17,3,18,2 X_5,13,6,12 X₆₇₁₈ X_9,16,10,17 X_15,20,16,21 X_13,22,14,7 X_21,14,22,15 X_19,4,20,5
Gauss code	{1, 4, -3, 11, -5, -6}, {6, -1, -7, 3, -2, 5, -9, 10, -8, 7, -4, 2, -11, 8, -10, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 2 + v u 2 + v 2 u -3 v u + u + v -1$ (db)
Jones polynomial	$-\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{2}{q^{13/2}}+\frac{2}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 3 a 7 + 2 z a 7 - z 5 a 5 -4 z 3 a 5 -4 z a 5 + z 3 a 3 + 2 z a 3 + a 3 z -1 - z a - a z -1$ (db)
Kauffman polynomial	$- z 7 a 9 + 5 z 5 a 9 -6 z 3 a 9 + z a 9 -2 z 8 a 8 + 11 z 6 a 8 -17 z 4 a 8 + 8 z 2 a 8 - z 9 a 7 + 4 z 7 a 7 -2 z 5 a 7 - z 3 a 7 - z a 7 -3 z 8 a 6 + 16 z 6 a 6 -24 z 4 a 6 + 12 z 2 a 6 - z 9 a 5 + 5 z 7 a 5 -8 z 5 a 5 + 9 z 3 a 5 -5 z a 5 - z 8 a 4 + 5 z 6 a 4 -7 z 4 a 4 + 4 z 2 a 4 - z 5 a 3 + 4 z 3 a 3 -4 z a 3 + a 3 z -1 - a 2 - z a + a z -1$ (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 L11n138. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -8$	${\mathbb Z}$
$r = -7$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -3$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}$
$r = -1$	${\mathbb Z}_2$	${\mathbb Z}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$