L10a48

(Knotscape image)

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

Visit L10a48's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-3 v u + 4 u + 4 v -3$ (db)
Jones polynomial	$-q^{11/2}+2 q^{9/2}-2 q^{7/2}+3 q^{5/2}-4 q^{3/2}+4 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 5 z -1 -2 z a 3 - a 3 z -1 + z 3 a + z 3 a -1 + z 3 a -3 + z a -3 - z a -5$ (db)
Kauffman polynomial	$- z 9 a -1 - z 9 a -3 -3 z 8 a -2 -2 z 8 a -4 - z 8 - a z 7 + 4 z 7 a -1 + 4 z 7 a -3 - z 7 a -5 - a 2 z 6 + 14 z 6 a -2 + 11 z 6 a -4 + 2 z 6 - a 3 z 5 + a z 5 -6 z 5 a -1 -3 z 5 a -3 + 5 z 5 a -5 - a 4 z 4 -18 z 4 a -2 -17 z 4 a -4 - a 5 z 3 - a 3 z 3 + 6 z 3 a -1 -6 z 3 a -5 + 7 z 2 a -2 + 7 z 2 a -4 + 2 a 5 z + 2 a 3 z - z a -1 + z a -5 + a 4 - a 5 z -1 - a 3 z -1$ (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 L10a48. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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