L10a122

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{u v w^3-3 u v w^2+3 u v w-u v-u w^3+4 u w^2-6 u w+2 u-2 v w^3+6 v w^2-4 v w+v+w^3-3 w^2+3 w-1}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db)
Jones polynomial	$q 4 -4 q 3 + 9 q 2 -11 q + 14-14 q -1 + 13 q -2 -9 q -3 + 6 q -4 -2 q -5 + q -6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a 6 z -2 + a 6 -3 z 2 a 4 -3 a 4 z -2 -5 a 4 + 3 z 4 a 2 + 7 z 2 a 2 + 4 a 2 z -2 + 8 a 2 - z 6 -3 z 4 -6 z 2 -3 z -2 -6 + z 4 a -2 + z 2 a -2 + a -2 z -2 + 2 a -2$ (db)
Kauffman polynomial	$a 3 z 9 + a z 9 + 2 a 4 z 8 + 7 a 2 z 8 + 5 z 8 + 2 a 5 z 7 + 7 a 3 z 7 + 15 a z 7 + 10 z 7 a -1 + a 6 z 6 + a 4 z 6 -2 a 2 z 6 + 9 z 6 a -2 + 7 z 6 -5 a 5 z 5 -20 a 3 z 5 -32 a z 5 -13 z 5 a -1 + 4 z 5 a -3 -4 a 6 z 4 -16 a 4 z 4 -30 a 2 z 4 -11 z 4 a -2 + z 4 a -4 -30 z 4 + 3 a 5 z 3 + 12 a 3 z 3 + 14 a z 3 + 4 z 3 a -1 - z 3 a -3 + 6 a 6 z 2 + 24 a 4 z 2 + 40 a 2 z 2 + 6 z 2 a -2 + 28 z 2 + a 5 z + a 3 z + a z + z a -1 -4 a 6 -14 a 4 -21 a 2 -4 a -2 -14- a 5 z -1 - a 3 z -1 - a z -1 - a -1 z -1 + a 6 z -2 + 3 a 4 z -2 + 4 a 2 z -2 + a -2 z -2 + 3 z -2$ (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 =

0 is the signature of L10a122. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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