L10a170

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,6,13,5 X₈₄₉₃ X_2,16,3,15 X_16,7,17,8 X_14,19,11,20 X_20,13,15,14 X_18,10,19,9 X_10,12,5,11 X_4,17,1,18
Gauss code	{1, -4, 3, -10}, {9, -2, 7, -6}, {2, -1, 5, -3, 8, -9}, {4, -5, 10, -8, 6, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 2 + v u 2 + v 2 w u 2 -2 v w u 2 + w u 2 + v 2 x u 2 - v x u 2 - v 2 w x u 2 + 2 v w x u 2 - w x u 2 + 2 v 2 u -3 v u - v 2 w u + 3 v w u -2 w u -2 v 2 x u + 3 v x u + v 2 w x u -3 v w x u + 2 w x u - x u + u - v 2 + 2 v - v w + w + v 2 x -2 v x + v w x - w x + x -1$ (db)
Jones polynomial	$q^{11/2}-4 q^{9/2}+8 q^{7/2}-13 q^{5/2}+14 q^{3/2}-18 \sqrt{q}+\frac{13}{\sqrt{q}}-\frac{13}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 2 a z 5 -4 z 5 a -1 + z 5 a -3 - a 3 z 3 + 5 a z 3 -6 z 3 a -1 + 2 z 3 a -3 - a 3 z + 2 a z -2 z a -1 + z a -3 + a 3 z -1 -3 a z -1 + 3 a -1 z -1 - a -3 z -1 + a 3 z -3 -3 a z -3 + 3 a -1 z -3 - a -3 z -3$ (db)
Kauffman polynomial	$-2 a z 9 -2 z 9 a -1 -5 a 2 z 8 -8 z 8 a -2 -13 z 8 -4 a 3 z 7 -11 a z 7 -18 z 7 a -1 -11 z 7 a -3 - a 4 z 6 + 9 a 2 z 6 + 6 z 6 a -2 -8 z 6 a -4 + 24 z 6 + 11 a 3 z 5 + 43 a z 5 + 53 z 5 a -1 + 17 z 5 a -3 -4 z 5 a -5 + 2 a 4 z 4 + a 2 z 4 + 6 z 4 a -2 + 7 z 4 a -4 - z 4 a -6 -3 z 4 -10 a 3 z 3 -40 a z 3 -44 z 3 a -1 -12 z 3 a -3 + 2 z 3 a -5 - a 4 z 2 -3 a 2 z 2 -3 z 2 a -2 - z 2 a -4 -4 z 2 + 3 a 3 z + 11 a z + 11 z a -1 + 3 z a -3 -4 a 2 -4 a -2 -7 + 2 a 3 z -1 + 3 a z -1 + 3 a -1 z -1 + 2 a -3 z -1 + 3 a 2 z -2 + 3 a -2 z -2 + 6 z -2 - a 3 z -3 -3 a z -3 -3 a -1 z -3 - a -3 z -3$ (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 =

1 is the signature of L10a170. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{6}$
$r = -1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{10}$
$r = 1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{8}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$