L10a168

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_14,7,15,8 X_8,13,5,14 X_18,12,19,11 X_20,16,17,15 X_16,20,9,19 X_12,18,13,17 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -4}, {8, -5, 7, -6}, {10, -2, 5, -8, 4, -3, 6, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 3 + w u 3 - u 3 -3 v u 2 + 2 v w u 2 -3 w u 2 + 2 v x u 2 - v w x u 2 + 2 w x u 2 -2 x u 2 + 2 u 2 + 2 v u -2 v w u + 2 w u -3 v x u + 2 v w x u -3 w x u + 2 x u - u + v x - v w x + w x$ (db)
Jones polynomial	$-q^{7/2}+3 q^{5/2}-8 q^{3/2}+10 \sqrt{q}-\frac{14}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{14}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 7 z -1 -3 z a 5 - a 5 z -1 + a 5 z -3 + 3 z 3 a 3 -4 a 3 z -1 -3 a 3 z -3 - z 5 a + z 3 a + 5 z a + 7 a z -1 + 3 a z -3 + 2 z 3 a -1 - z a -1 -3 a -1 z -1 - a -1 z -3 - z a -3$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 -3 a 4 z 8 -7 a 2 z 8 -4 z 8 -3 a 5 z 7 -10 a 3 z 7 -13 a z 7 -6 z 7 a -1 -2 a 6 z 6 + 3 a 2 z 6 -3 z 6 a -2 -2 z 6 - a 7 z 5 + 3 a 5 z 5 + 27 a 3 z 5 + 37 a z 5 + 13 z 5 a -1 - z 5 a -3 + 3 a 6 z 4 + 8 a 4 z 4 + 21 a 2 z 4 + 4 z 4 a -2 + 20 z 4 + 3 a 7 z 3 - a 5 z 3 -32 a 3 z 3 -45 a z 3 -15 z 3 a -1 + 2 z 3 a -3 -15 a 4 z 2 -36 a 2 z 2 - z 2 a -2 -22 z 2 -3 a 7 z + 3 a 5 z + 24 a 3 z + 33 a z + 14 z a -1 - z a -3 - a 6 + 11 a 4 + 24 a 2 + 13 + a 7 z -1 -3 a 5 z -1 -12 a 3 z -1 -14 a z -1 -6 a -1 z -1 -3 a 4 z -2 -6 a 2 z -2 -3 z -2 + a 5 z -3 + 3 a 3 z -3 + 3 a z -3 + a -1 z -3$ (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 L10a168. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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