L10a158

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 3 - v w u 3 + w u 3 - u 3 + v 2 u 2 - v w 2 u 2 + w 2 u 2 -2 v u 2 - v 2 w u 2 + 4 v w u 2 -2 w u 2 + u 2 - v 2 u - v 2 w 2 u + 2 v w 2 u - w 2 u + v u + 2 v 2 w u -4 v w u + w u + v 2 w 2 - v w 2 - v 2 w + v w$ (db)
Jones polynomial	$- q 5 + 3 q 4 -5 q 3 + 9 q 2 -10 q + 12-10 q -1 + 9 q -2 -5 q -3 + 3 q -4 - q -5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 2 a 4 + z 4 a 2 + a 2 z -2 + a 2 + 2 z 4 + z 2 -2 z -2 -2 + z 4 a -2 + a -2 z -2 + a -2 - z 2 a -4$ (db)
Kauffman polynomial	$2 a z 9 + 2 z 9 a -1 + 4 a 2 z 8 + 4 z 8 a -2 + 8 z 8 + 4 a 3 z 7 - a z 7 - z 7 a -1 + 4 z 7 a -3 + 3 a 4 z 6 -9 a 2 z 6 -9 z 6 a -2 + 3 z 6 a -4 -24 z 6 + a 5 z 5 -8 a 3 z 5 -2 a z 5 -2 z 5 a -1 -8 z 5 a -3 + z 5 a -5 -7 a 4 z 4 + 9 a 2 z 4 + 9 z 4 a -2 -7 z 4 a -4 + 32 z 4 -2 a 5 z 3 + 3 a 3 z 3 + 3 a z 3 + 3 z 3 a -1 + 3 z 3 a -3 -2 z 3 a -5 + 3 a 4 z 2 -3 a 2 z 2 -3 z 2 a -2 + 3 z 2 a -4 -12 z 2 + 2 a z + 2 z a -1 -2 a 2 -2 a -2 -3-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 =

0 is the signature of L10a158. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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