L10a166

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 2 - v u 2 - v 2 w u 2 + v w u 2 - v 2 x u 2 + 2 v x u 2 - v w x u 2 - v 2 u + 2 v u + 2 v 2 w u -3 v w u + w u + v 2 x u -3 v x u - v 2 w x u + 2 v w x u - w x u + 2 x u - u - v + 2 v w - w + v x - v w x + w x - x$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{13}{q^{13/2}}+\frac{11}{q^{15/2}}-\frac{12}{q^{17/2}}+\frac{8}{q^{19/2}}-\frac{6}{q^{21/2}}+\frac{2}{q^{23/2}}-\frac{1}{q^{25/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a 13 z -1 + a 13 z -3 -4 z a 11 -8 a 11 z -1 -3 a 11 z -3 + 6 z 3 a 9 + 17 z a 9 + 13 a 9 z -1 + 3 a 9 z -3 -3 z 5 a 7 -11 z 3 a 7 -13 z a 7 -6 a 7 z -1 - a 7 z -3 - z 5 a 5 -2 z 3 a 5$ (db)
Kauffman polynomial	$- z 5 a 15 + 3 z 3 a 15 -3 z a 15 + a 15 z -1 -2 z 6 a 14 + 3 z 4 a 14 - a 14 -3 z 7 a 13 + 4 z 5 a 13 -2 z 3 a 13 + 3 z a 13 -3 a 13 z -1 + a 13 z -3 -2 z 8 a 12 -4 z 6 a 12 + 15 z 4 a 12 -16 z 2 a 12 -3 a 12 z -2 + 11 a 12 - z 9 a 11 -6 z 7 a 11 + 15 z 5 a 11 -20 z 3 a 11 + 21 z a 11 -12 a 11 z -1 + 3 a 11 z -3 -6 z 8 a 10 + 5 z 6 a 10 + 14 z 4 a 10 -33 z 2 a 10 -6 a 10 z -2 + 24 a 10 - z 9 a 9 -9 z 7 a 9 + 28 z 5 a 9 -37 z 3 a 9 + 28 z a 9 -14 a 9 z -1 + 3 a 9 z -3 -4 z 8 a 8 + 4 z 6 a 8 + 7 z 4 a 8 -17 z 2 a 8 -3 a 8 z -2 + 13 a 8 -6 z 7 a 7 + 17 z 5 a 7 -20 z 3 a 7 + 13 z a 7 -6 a 7 z -1 + a 7 z -3 -3 z 6 a 6 + 5 z 4 a 6 - z 5 a 5 + 2 z 3 a 5$ (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 =

-5 is the signature of L10a166. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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