L11n164

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_12,17,13,18 X_14,5,15,6 X_4,13,5,14 X_18,11,19,12 X_19,7,20,22 X_15,21,16,20 X_21,17,22,16 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, 6, -3, 5, -4, -8, 9, 3, -6, -7, 8, -9, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 4 + u 4 - v 2 u 3 + 2 v u 3 -2 u 3 + v 2 u 2 -3 v u 2 + u 2 -2 v 2 u + 2 v u - u + v 2 - v$ (db)
Jones polynomial	$\sqrt{q}-\frac{3}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{6}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{2}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 3 a 7 + 2 z a 7 + a 7 z -1 - z 5 a 5 -3 z 3 a 5 -3 z a 5 - a 5 z -1 - z 5 a 3 -3 z 3 a 3 -3 z a 3 + z 3 a + z a$ (db)
Kauffman polynomial	$- z 7 a 9 + 5 z 5 a 9 -8 z 3 a 9 + 4 z a 9 -2 z 8 a 8 + 9 z 6 a 8 -11 z 4 a 8 + 3 z 2 a 8 - z 9 a 7 + z 7 a 7 + 8 z 5 a 7 -10 z 3 a 7 + a 7 z -1 -4 z 8 a 6 + 14 z 6 a 6 -11 z 4 a 6 + 3 z 2 a 6 - a 6 - z 9 a 5 + 7 z 5 a 5 -3 z 3 a 5 -2 z a 5 + a 5 z -1 -2 z 8 a 4 + 4 z 6 a 4 -2 z 7 a 3 + 4 z 5 a 3 -4 z 3 a 3 + 3 z a 3 - z 6 a 2 - z 2 a 2 -3 z 3 a + z a - z 2$ (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 L11n164. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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