L10a159

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v w 2 u 3 - v 2 w u 3 + v w u 3 - v 2 u 2 - v 2 w 2 u 2 + 2 v w 2 u 2 - w 2 u 2 + v u 2 + 2 v 2 w u 2 -4 v w u 2 + w u 2 + v 2 u - v w 2 u + w 2 u -2 v u - v 2 w u + 4 v w u -2 w u + u + v - v w + w$ (db)
Jones polynomial	$q -3 -2 q -4 + 5 q -5 -7 q -6 + 10 q -7 -10 q -8 + 11 q -9 -8 q -10 + 6 q -11 -3 q -12 + q -13$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$z 2 a 12 + a 12 z -2 + 2 a 12 -3 z 4 a 10 -10 z 2 a 10 -2 a 10 z -2 -9 a 10 + 2 z 6 a 8 + 9 z 4 a 8 + 13 z 2 a 8 + a 8 z -2 + 7 a 8 + z 6 a 6 + 4 z 4 a 6 + 4 z 2 a 6$ (db)
Kauffman polynomial	$z 4 a 16 - z 2 a 16 + 3 z 5 a 15 -3 z 3 a 15 + 5 z 6 a 14 -7 z 4 a 14 + 5 z 2 a 14 -2 a 14 + 5 z 7 a 13 -6 z 5 a 13 + 3 z 3 a 13 + 3 z 8 a 12 + z 6 a 12 -8 z 4 a 12 + 5 z 2 a 12 - a 12 z -2 + 3 a 12 + z 9 a 11 + 6 z 7 a 11 -17 z 5 a 11 + 17 z 3 a 11 -9 z a 11 + 2 a 11 z -1 + 6 z 8 a 10 -16 z 6 a 10 + 22 z 4 a 10 -23 z 2 a 10 -2 a 10 z -2 + 11 a 10 + z 9 a 9 + 3 z 7 a 9 -14 z 5 a 9 + 15 z 3 a 9 -9 z a 9 + 2 a 9 z -1 + 3 z 8 a 8 -11 z 6 a 8 + 18 z 4 a 8 -18 z 2 a 8 - a 8 z -2 + 7 a 8 + 2 z 7 a 7 -6 z 5 a 7 + 4 z 3 a 7 + z 6 a 6 -4 z 4 a 6 + 4 z 2 a 6$ (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 =

-6 is the signature of L10a159. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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