L10a164

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_14,3,15,4 X_12,15,7,16 X_10,19,11,20 X_16,9,17,10 X_20,11,13,12 X_18,5,19,6 X₂₇₃₈ X_4,13,5,14 X_6,17,1,18
Gauss code	{1, -8, 2, -9, 7, -10}, {8, -1, 5, -4, 6, -3}, {9, -2, 3, -5, 10, -7, 4, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 2 v 3 + u v 3 - u w v 3 + 2 u 2 v 2 -2 u w 2 v 2 + w 2 v 2 -3 u v 2 -2 u 2 w v 2 + 4 u w v 2 -2 w v 2 + v 2 - u 2 v - u 2 w 2 v + 3 u w 2 v -2 w 2 v + 2 u v + 2 u 2 w v -4 u w v + 2 w v - u w 2 + w 2 + u w$ (db)
Jones polynomial	$q -2 -3 q -3 + 7 q -4 -10 q -5 + 13 q -6 -13 q -7 + 13 q -8 -9 q -9 + 7 q -10 -3 q -11 + q -12$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a 12 z -2 + a 12 -4 z 2 a 10 -2 a 10 z -2 -6 a 10 + 3 z 4 a 8 + 6 z 2 a 8 + a 8 z -2 + 4 a 8 + 3 z 4 a 6 + 5 z 2 a 6 + a 6 + z 4 a 4 + z 2 a 4$ (db)
Kauffman polynomial	$z 6 a 14 -3 z 4 a 14 + 3 z 2 a 14 - a 14 + 3 z 7 a 13 -8 z 5 a 13 + 5 z 3 a 13 + 4 z 8 a 12 -9 z 6 a 12 + 4 z 4 a 12 -3 z 2 a 12 - a 12 z -2 + 5 a 12 + 2 z 9 a 11 + 3 z 7 a 11 -18 z 5 a 11 + 17 z 3 a 11 -9 z a 11 + 2 a 11 z -1 + 10 z 8 a 10 -25 z 6 a 10 + 26 z 4 a 10 -23 z 2 a 10 -2 a 10 z -2 + 11 a 10 + 2 z 9 a 9 + 7 z 7 a 9 -22 z 5 a 9 + 19 z 3 a 9 -9 z a 9 + 2 a 9 z -1 + 6 z 8 a 8 -9 z 6 a 8 + 10 z 4 a 8 -10 z 2 a 8 - a 8 z -2 + 5 a 8 + 7 z 7 a 7 -9 z 5 a 7 + 5 z 3 a 7 + 6 z 6 a 6 -8 z 4 a 6 + 6 z 2 a 6 - a 6 + 3 z 5 a 5 -2 z 3 a 5 + z 4 a 4 - z 2 a 4$ (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 =

-4 is the signature of L10a164. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -5$	$i = -3$
$r = -10$	${\mathbb Z}$
$r = -9$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -8$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = -7$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -6$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{7}$
$r = -5$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -4$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -3$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -1$	${\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$