L10a167

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 2 + 2 v 2 w u 2 - v w u 2 + v 2 x u 2 - v x u 2 - v 2 w x u 2 + v w x u 2 + v 2 u - v u - v 2 w u + 2 v w u - w u - v 2 x u + 2 v x u - v w x u + w x u - x u + v - v w + w - v x - w x + 2 x -1$ (db)
Jones polynomial	$-q^{15/2}+3 q^{13/2}-6 q^{11/2}+7 q^{9/2}-10 q^{7/2}+8 q^{5/2}-9 q^{3/2}+5 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 -2 z 5 a -1 + 5 z 5 a -3 - z 5 a -5 + a z 3 -9 z 3 a -1 + 10 z 3 a -3 -3 z 3 a -5 + 4 a z -14 z a -1 + 13 z a -3 -3 z a -5 + 4 a z -1 -11 a -1 z -1 + 10 a -3 z -1 -3 a -5 z -1 + a z -3 -3 a -1 z -3 + 3 a -3 z -3 - a -5 z -3$ (db)
Kauffman polynomial	$- z 9 a -1 - z 9 a -3 -5 z 8 a -2 -4 z 8 a -4 - z 8 - a z 7 -6 z 7 a -3 -7 z 7 a -5 + 13 z 6 a -2 + 4 z 6 a -4 -7 z 6 a -6 + 2 z 6 + 6 a z 5 + 14 z 5 a -1 + 27 z 5 a -3 + 13 z 5 a -5 -6 z 5 a -7 + 6 z 4 a -2 + 11 z 4 a -4 + 8 z 4 a -6 -3 z 4 a -8 + 6 z 4 -13 a z 3 -30 z 3 a -1 -30 z 3 a -3 -6 z 3 a -5 + 6 z 3 a -7 - z 3 a -9 -33 z 2 a -2 -16 z 2 a -4 -17 z 2 + 13 a z + 28 z a -1 + 21 z a -3 + 3 z a -5 -3 z a -7 + 24 a -2 + 11 a -4 - a -6 + 13-6 a z -1 -14 a -1 z -1 -12 a -3 z -1 -3 a -5 z -1 + a -7 z -1 -6 a -2 z -2 -3 a -4 z -2 -3 z -2 + a z -3 + 3 a -1 z -3 + 3 a -3 z -3 + a -5 z -3$ (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 =

3 is the signature of L10a167. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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