L10a4

(Knotscape image)

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

Visit L10a4's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_12,6,13,5 X₈₄₉₃ X_16,10,17,9 X_20,18,5,17 X_18,11,19,12 X_10,19,11,20 X_2,14,3,13
Gauss code	{1, -10, 5, -3}, {4, -1, 2, -5, 6, -9, 8, -4, 10, -2, 3, -6, 7, -8, 9, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 4 v u 4 -4 u 4 -6 v u 3 + 6 u 3 + 6 v u 2 -6 u 2 -4 v u + 4 u + v -1$ (db)
Jones polynomial	$q^{11/2}-4 q^{9/2}+7 q^{7/2}-11 q^{5/2}+14 q^{3/2}-15 \sqrt{q}+\frac{13}{\sqrt{q}}-\frac{11}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 2 a z 5 -4 z 5 a -1 + z 5 a -3 - a 3 z 3 + 5 a z 3 -5 z 3 a -1 + 2 z 3 a -3 - a 3 z + a z + a 3 z -1 -2 a z -1 + 2 a -1 z -1 - a -3 z -1$ (db)
Kauffman polynomial	$-2 a z 9 -2 z 9 a -1 -5 a 2 z 8 -6 z 8 a -2 -11 z 8 -4 a 3 z 7 -7 a z 7 -11 z 7 a -1 -8 z 7 a -3 - a 4 z 6 + 11 a 2 z 6 + 2 z 6 a -2 -7 z 6 a -4 + 21 z 6 + 11 a 3 z 5 + 29 a z 5 + 30 z 5 a -1 + 8 z 5 a -3 -4 z 5 a -5 + 2 a 4 z 4 -3 a 2 z 4 + 9 z 4 a -2 + 7 z 4 a -4 - z 4 a -6 -4 z 4 -8 a 3 z 3 -21 a z 3 -17 z 3 a -1 - z 3 a -3 + 3 z 3 a -5 - a 4 z 2 -2 a 2 z 2 -6 z 2 a -2 -2 z 2 a -4 -5 z 2 + a 3 z -2 z a -1 - z a -3 + 1 + a 3 z -1 + 2 a z -1 + 2 a -1 z -1 + a -3 z -1$ (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 =

1 is the signature of L10a4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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