L11n156

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_11,21,12,20 X_5,12,6,13 X_19,4,20,5 X_14,18,15,17 X_16,8,17,7 X_22,16,7,15 X_18,14,19,13 X_6,21,1,22
Gauss code	{1, -2, 3, 6, -5, -11}, {8, -1, 2, -3, -4, 5, 10, -7, 9, -8, 7, -10, -6, 4, 11, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 4 - v 2 u 3 + 4 v u 3 -2 u 3 + 2 v 2 u 2 -5 v u 2 + 2 u 2 -2 v 2 u + 4 v u - u -2 v$ (db)
Jones polynomial	$2 q^{5/2}-4 q^{3/2}+6 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{9}{q^{3/2}}-\frac{9}{q^{5/2}}+\frac{7}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 3 a 5 + 2 z a 5 + a 5 z -1 - z 5 a 3 -2 z 3 a 3 - z a 3 -2 z 5 a -7 z 3 a -8 z a -2 a z -1 + 2 z 3 a -1 + 4 z a -1 + a -1 z -1$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 -2 a 4 z 8 -4 a 2 z 8 -2 z 8 -3 a 5 z 7 - a 3 z 7 + a z 7 - z 7 a -1 -2 a 6 z 6 + a 4 z 6 + 9 a 2 z 6 + 6 z 6 - a 7 z 5 + 7 a 5 z 5 + 2 a 3 z 5 -7 a z 5 - z 5 a -1 + 4 a 6 z 4 + 3 a 4 z 4 -14 a 2 z 4 -3 z 4 a -2 -16 z 4 + 3 a 7 z 3 -7 a 5 z 3 -2 a 3 z 3 + 13 a z 3 + 5 z 3 a -1 - a 6 z 2 -3 a 4 z 2 + 10 a 2 z 2 + 6 z 2 a -2 + 18 z 2 -2 a 7 z + 5 a 5 z + a 3 z -10 a z -4 z a -1 + a 4 -3 a 2 -2 a -2 -5- a 5 z -1 + 2 a z -1 + a -1 z -1$ (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 L11n156. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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