L10a80

(Knotscape image)

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

Visit L10a80's page at the original Knot Atlas.

Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_18,10,19,9 X_20,13,7,14 X_14,19,15,20 X_10,16,11,15 X_16,6,17,5 X₂₇₃₈ X_4,11,5,12 X_6,18,1,17
Gauss code	{1, -8, 2, -9, 7, -10}, {8, -1, 3, -6, 9, -2, 4, -5, 6, -7, 10, -3, 5, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 4 + u 4 -2 v 2 u 3 + 4 v u 3 -3 u 3 + 4 v 2 u 2 -5 v u 2 + 4 u 2 -3 v 2 u + 4 v u -2 u + v 2 - v$ (db)
Jones polynomial	$-q^{9/2}+3 q^{7/2}-6 q^{5/2}+9 q^{3/2}-11 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{11}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z a 5 + a 5 z -1 -3 z 3 a 3 -6 z a 3 -2 a 3 z -1 + 2 z 5 a + 6 z 3 a + 6 z a + 2 a z -1 + z 5 a -1 + z 3 a -1 - z a -1 - a -1 z -1 - z 3 a -3 - z a -3$ (db)
Kauffman polynomial	$-2 a 3 z 9 -2 a z 9 -3 a 4 z 8 -10 a 2 z 8 -7 z 8 - a 5 z 7 + a 3 z 7 -8 a z 7 -10 z 7 a -1 + 12 a 4 z 6 + 34 a 2 z 6 -9 z 6 a -2 + 13 z 6 + 4 a 5 z 5 + 19 a 3 z 5 + 42 a z 5 + 21 z 5 a -1 -6 z 5 a -3 -14 a 4 z 4 -30 a 2 z 4 + 13 z 4 a -2 -3 z 4 a -4 -6 a 5 z 3 -32 a 3 z 3 -44 a z 3 -13 z 3 a -1 + 4 z 3 a -3 - z 3 a -5 + 4 a 4 z 2 + 7 a 2 z 2 -5 z 2 a -2 -2 z 2 + 4 a 5 z + 15 a 3 z + 17 a z + 5 z a -1 - z a -3 - a 2 - a 5 z -1 -2 a 3 z -1 -2 a z -1 - a -1 z -1$ (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 L10a80. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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