L10a169

(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_14,9,11,10 X_20,13,15,14 X_10,19,5,20 X_18,12,19,11 X_4,17,1,18
Gauss code	{1, -4, 3, -10}, {9, -2, 7, -6}, {2, -1, 5, -3, 6, -8}, {4, -5, 10, -9, 8, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 2 + 2 v u 2 + v 2 w u 2 -2 v w u 2 + w u 2 + v 2 x u 2 -2 v x u 2 - v 2 w x u 2 + 2 v w x u 2 - w x u 2 + x u 2 - u 2 + 2 v 2 u -4 v u -2 v 2 w u + 4 v w u -2 w u -2 v 2 x u + 4 v x u + 2 v 2 w x u -4 v w x u + 2 w x u -2 x u + 2 u - v 2 + 2 v + v 2 w -2 v w + w + v 2 x -2 v x - v 2 w x + 2 v w x - w x + x -1$ (db)
Jones polynomial	$-q^{7/2}+5 q^{5/2}-11 q^{3/2}+15 \sqrt{q}-\frac{22}{\sqrt{q}}+\frac{20}{q^{3/2}}-\frac{22}{q^{5/2}}+\frac{15}{q^{7/2}}-\frac{11}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 -2 a 3 z 5 + 3 a z 5 - z 5 a -1 + a 5 z 3 -3 a 3 z 3 + 3 a z 3 - z 3 a -1 - a 3 z -1 + 2 a z -1 - a -1 z -1 + a 5 z -3 -3 a 3 z -3 + 3 a z -3 - a -1 z -3$ (db)
Kauffman polynomial	$-4 a 3 z 9 -4 a z 9 -11 a 4 z 8 -22 a 2 z 8 -11 z 8 -11 a 5 z 7 -19 a 3 z 7 -19 a z 7 -11 z 7 a -1 -5 a 6 z 6 + 12 a 4 z 6 + 34 a 2 z 6 -5 z 6 a -2 + 12 z 6 - a 7 z 5 + 17 a 5 z 5 + 48 a 3 z 5 + 48 a z 5 + 17 z 5 a -1 - z 5 a -3 + 4 a 6 z 4 -8 a 2 z 4 + 4 z 4 a -2 -8 a 5 z 3 -24 a 3 z 3 -24 a z 3 -8 z 3 a -1 - a 3 z - a z + 2 a 4 + 3 a 2 + 2- a 5 z -1 - a -1 z -1 -3 a 4 z -2 -6 a 2 z -2 -3 z -2 + a 5 z -3 + 3 a 3 z -3 + 3 a z -3 + a -1 z -3$ (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 L10a169. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -3$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -2$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{12}$
$r = -1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{14}$
$r = 1$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$