L11a138

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_2,11,3,12 X_14,3,15,4 X_12,5,13,6 X_22,13,5,14 X_4,21,1,22 X_20,16,21,15 X_16,8,17,7 X_8,20,9,19 X_18,10,19,9 X_10,18,11,17
Gauss code	{1, -2, 3, -6}, {4, -1, 8, -9, 10, -11, 2, -4, 5, -3, 7, -8, 11, -10, 9, -7, 6, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-4 v u 3 + 4 u 3 + 12 v u 2 -13 u 2 -13 v u + 12 u + 4 v -4$ (db)
Jones polynomial	$q^{9/2}-4 q^{7/2}+8 q^{5/2}-13 q^{3/2}+18 \sqrt{q}-\frac{21}{\sqrt{q}}+\frac{21}{q^{3/2}}-\frac{19}{q^{5/2}}+\frac{13}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 3 a 5 + a 5 z -1 - z 5 a 3 - z a 3 - a 3 z -1 -2 z 5 a -2 z 3 a - z a - z 5 a -1 + z 3 a -3$ (db)
Kauffman polynomial	$-2 a 2 z 10 -2 z 10 -6 a 3 z 9 -12 a z 9 -6 z 9 a -1 -9 a 4 z 8 -12 a 2 z 8 -7 z 8 a -2 -10 z 8 -8 a 5 z 7 + 2 a 3 z 7 + 25 a z 7 + 11 z 7 a -1 -4 z 7 a -3 -4 a 6 z 6 + 15 a 4 z 6 + 35 a 2 z 6 + 19 z 6 a -2 - z 6 a -4 + 36 z 6 - a 7 z 5 + 14 a 5 z 5 + 10 a 3 z 5 -18 a z 5 -3 z 5 a -1 + 10 z 5 a -3 + 5 a 6 z 4 -10 a 4 z 4 -34 a 2 z 4 -15 z 4 a -2 + 2 z 4 a -4 -36 z 4 + a 7 z 3 -8 a 5 z 3 -9 a 3 z 3 + 4 a z 3 - z 3 a -1 -5 z 3 a -3 + 3 a 4 z 2 + 11 a 2 z 2 + 4 z 2 a -2 + 12 z 2 + 3 a 5 z + 2 a 3 z - a z + a 4 - a 5 z -1 - a 3 z -1$ (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 =

-1 is the signature of L11a138. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{4}$
$r = -3$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 0$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{11}$
$r = 1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 2$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$