L10a5

(Knotscape image)

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

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Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_4,19,1,20 X_14,10,15,9 X₈₄₉₃ X_12,5,13,6 X_20,13,5,14 X_16,12,17,11 X_10,16,11,15 X_2,18,3,17
Gauss code	{1, -10, 5, -3}, {6, -1, 2, -5, 4, -9, 8, -6, 7, -4, 9, -8, 10, -2, 3, -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	$q^{11/2}-4 q^{9/2}+8 q^{7/2}-12 q^{5/2}+14 q^{3/2}-15 \sqrt{q}+\frac{13}{\sqrt{q}}-\frac{11}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 2 a z 5 -4 z 5 a -1 + z 5 a -3 - a 3 z 3 + 6 a z 3 -6 z 3 a -1 + 2 z 3 a -3 -2 a 3 z + 5 a z -4 z a -1 + z a -3 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 a z 9 -2 z 9 a -1 -4 a 2 z 8 -7 z 8 a -2 -11 z 8 -3 a 3 z 7 -5 a z 7 -12 z 7 a -1 -10 z 7 a -3 - a 4 z 6 + 9 a 2 z 6 + 5 z 6 a -2 -8 z 6 a -4 + 23 z 6 + 9 a 3 z 5 + 25 a z 5 + 34 z 5 a -1 + 14 z 5 a -3 -4 z 5 a -5 + 3 a 4 z 4 -4 a 2 z 4 + 5 z 4 a -2 + 8 z 4 a -4 - z 4 a -6 -11 z 4 -9 a 3 z 3 -26 a z 3 -26 z 3 a -1 -7 z 3 a -3 + 2 z 3 a -5 -2 a 4 z 2 - a 2 z 2 -3 z 2 a -2 -2 z 2 a -4 + 4 a 3 z + 10 a z + 8 z a -1 + 2 z a -3 + 1- a z -1 - a -1 z -1$ (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 L10a5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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