L10a157

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 3 - v u 3 - v 2 w u 3 + v w u 3 - v 2 u 2 - v 2 w 2 u 2 + v w 2 u 2 + 2 v u 2 + 2 v 2 w u 2 -2 v w u 2 + w u 2 - u 2 + v 2 w 2 u -2 v w 2 u + w 2 u - v u - v 2 w u + 2 v w u -2 w u + u + v w 2 - w 2 - v w + w$ (db)
Jones polynomial	$- q 4 + 3 q 3 -4 q 2 + 8 q -8 + 10 q -1 -9 q -2 + 8 q -3 -5 q -4 + 3 q -5 - q -6$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a 2 z 6 + z 6 - a 4 z 4 + 3 a 2 z 4 - z 4 a -2 + 3 z 4 -2 a 4 z 2 + 2 a 2 z 2 -2 z 2 a -2 + z 2 + a 2 + a -2 -2 + a 2 z -2 + a -2 z -2 -2 z -2$ (db)
Kauffman polynomial	$2 a z 9 + 2 z 9 a -1 + 5 a 2 z 8 + 3 z 8 a -2 + 8 z 8 + 7 a 3 z 7 + a z 7 -5 z 7 a -1 + z 7 a -3 + 7 a 4 z 6 -9 a 2 z 6 -14 z 6 a -2 -30 z 6 + 5 a 5 z 5 -13 a 3 z 5 -16 a z 5 -2 z 5 a -1 -4 z 5 a -3 + 3 a 6 z 4 -10 a 4 z 4 - a 2 z 4 + 20 z 4 a -2 + 32 z 4 + a 7 z 3 -3 a 5 z 3 + 5 a 3 z 3 + 11 a z 3 + 6 z 3 a -1 + 4 z 3 a -3 - a 6 z 2 + 4 a 4 z 2 + 3 a 2 z 2 -8 z 2 a -2 -10 z 2 + 2 a z + 2 z a -1 -2 a 2 -2 a -2 -3-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 =

-2 is the signature of L10a157. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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