L10a163

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_20,10,13,9 X_6,20,1,19 X_18,7,19,8 X_4,11,5,12 X_16,6,17,5 X_10,16,11,15 X_12,17,7,18 X_2,13,3,14 X_14,3,15,4
Gauss code	{1, -9, 10, -5, 6, -3}, {4, -1, 2, -7, 5, -8}, {9, -10, 7, -6, 8, -4, 3, -2}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 w 2 u 3 + v w 2 u 3 + v 2 w u 3 - v w u 3 + v 2 u 2 + 2 v 2 w 2 u 2 -3 v w 2 u 2 + w 2 u 2 -2 v u 2 -3 v 2 w u 2 + 5 v w u 2 -2 w u 2 + u 2 - v 2 u - v 2 w 2 u + 2 v w 2 u - w 2 u + 3 v u + 2 v 2 w u -5 v w u + 3 w u -2 u - v + v w - w + 1$ (db)
Jones polynomial	$q 3 -4 q 2 + 8 q -11 + 16 q -1 -15 q -2 + 16 q -3 -12 q -4 + 8 q -5 -4 q -6 + q -7$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- a 2 z 8 + a 4 z 6 -5 a 2 z 6 + z 6 + 3 a 4 z 4 -8 a 2 z 4 + 3 z 4 + 2 a 4 z 2 -4 a 2 z 2 + 2 z 2 - a 2 + 1 + a 4 z -2 -2 a 2 z -2 + z -2$ (db)
Kauffman polynomial	$3 a 3 z 9 + 3 a z 9 + 8 a 4 z 8 + 14 a 2 z 8 + 6 z 8 + 10 a 5 z 7 + 9 a 3 z 7 + 3 a z 7 + 4 z 7 a -1 + 8 a 6 z 6 -7 a 4 z 6 -31 a 2 z 6 + z 6 a -2 -15 z 6 + 4 a 7 z 5 -12 a 5 z 5 -26 a 3 z 5 -20 a z 5 -10 z 5 a -1 + a 8 z 4 -8 a 6 z 4 -3 a 4 z 4 + 18 a 2 z 4 -2 z 4 a -2 + 10 z 4 -2 a 7 z 3 + 4 a 5 z 3 + 12 a 3 z 3 + 12 a z 3 + 6 z 3 a -1 + 3 a 6 z 2 + 2 a 4 z 2 -4 a 2 z 2 + z 2 a -2 -2 z 2 - a 3 z - a z + a 4 + a 2 + 1 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 =

-2 is the signature of L10a163. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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