L10a22

(Knotscape image)

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

Visit L10a22's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_14,9,15,10 X_4,19,1,20 X_12,6,13,5 X_10,4,11,3 X_20,12,5,11 X_2,14,3,13 X_18,16,19,15 X_16,8,17,7 X_8,18,9,17
Gauss code	{1, -7, 5, -3}, {4, -1, 9, -10, 2, -5, 6, -4, 7, -2, 8, -9, 10, -8, 3, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 4 v u 4 -4 u 4 -7 v u 3 + 7 u 3 + 7 v u 2 -7 u 2 -4 v u + 4 u + v -1$ (db)
Jones polynomial	$q^{17/2}-4 q^{15/2}+9 q^{13/2}-12 q^{11/2}+15 q^{9/2}-17 q^{7/2}+14 q^{5/2}-12 q^{3/2}+7 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{1}{q^{3/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 - z 5 a -1 + 4 z 5 a -3 -2 z 5 a -5 -2 z 3 a -1 + 6 z 3 a -3 -5 z 3 a -5 + z 3 a -7 + 3 z a -3 -4 z a -5 + z a -7 + a -1 z -1 -2 a -5 z -1 + a -7 z -1$ (db)
Kauffman polynomial	$-2 z 9 a -3 -2 z 9 a -5 -5 z 8 a -2 -12 z 8 a -4 -7 z 8 a -6 -4 z 7 a -1 -9 z 7 a -3 -16 z 7 a -5 -11 z 7 a -7 + 10 z 6 a -2 + 19 z 6 a -4 - z 6 a -6 -9 z 6 a -8 - z 6 + 11 z 5 a -1 + 33 z 5 a -3 + 41 z 5 a -5 + 15 z 5 a -7 -4 z 5 a -9 -2 z 4 a -2 + 4 z 4 a -4 + 19 z 4 a -6 + 10 z 4 a -8 - z 4 a -10 + 2 z 4 -9 z 3 a -1 -26 z 3 a -3 -26 z 3 a -5 -8 z 3 a -7 + z 3 a -9 -3 z 2 a -2 -11 z 2 a -4 -14 z 2 a -6 -5 z 2 a -8 - z 2 + z a -1 + 6 z a -3 + 7 z a -5 + 2 z a -7 - a -2 + 3 a -4 + 5 a -6 + 2 a -8 + a -1 z -1 -2 a -5 z -1 - a -7 z -1$ (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 L10a22. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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