L11n161

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_16,7,17,8 X_10,4,11,3 X_2,15,3,16 X_14,10,15,9 X_18,11,19,12 X_5,13,6,12 X_6,21,1,22 X_13,20,14,21 X_22,17,7,18 X_19,4,20,5
Gauss code	{1, -4, 3, 11, -7, -8}, {2, -1, 5, -3, 6, 7, -9, -5, 4, -2, 10, -6, -11, 9, 8, -10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 2 + 3 v u 2 - u 2 + 2 v 2 u -5 v u + 2 u - v 2 + 3 v -2$ (db)
Jones polynomial	$\sqrt{q}-\frac{3}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{6}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 3 a 7 + z a 7 - a 7 z -1 - z 5 a 5 -2 z 3 a 5 + z a 5 + 3 a 5 z -1 - z 5 a 3 -3 z 3 a 3 -4 z a 3 -2 a 3 z -1 + z 3 a + z a$ (db)
Kauffman polynomial	$- z 7 a 9 + 4 z 5 a 9 -4 z 3 a 9 + z a 9 -3 z 8 a 8 + 14 z 6 a 8 -18 z 4 a 8 + 6 z 2 a 8 + a 8 -2 z 9 a 7 + 6 z 7 a 7 + z 5 a 7 -7 z 3 a 7 + 2 z a 7 - a 7 z -1 -7 z 8 a 6 + 30 z 6 a 6 -34 z 4 a 6 + 9 z 2 a 6 + 3 a 6 -2 z 9 a 5 + 4 z 7 a 5 + 6 z 5 a 5 -10 z 3 a 5 + 5 z a 5 -3 a 5 z -1 -4 z 8 a 4 + 15 z 6 a 4 -16 z 4 a 4 + 4 z 2 a 4 + 3 a 4 -3 z 7 a 3 + 9 z 5 a 3 -10 z 3 a 3 + 6 z a 3 -2 a 3 z -1 - z 6 a 2 -3 z 3 a + 2 z a - z 2$ (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 L11n161. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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