L10n78

(Knotscape image)

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

Visit L10n78's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_5,14,6,15 X₃₈₄₉ X_2,16,3,15 X_16,7,17,8 X_9,18,10,19 X_11,20,12,13 X_13,12,14,5 X_17,1,18,4 X_19,10,20,11
Gauss code	{1, -4, -3, 9}, {-2, -1, 5, 3, -6, 10, -7, 8}, {-8, 2, 4, -5, -9, 6, -10, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 w u 3 + v 3 u 2 - v 2 u 2 + v 2 w u 2 - v w u 2 + v 2 u - v u + v w u - w u + v$ (db)
Jones polynomial	$2 q -2 - q -3 + 3 q -4 -3 q -5 + 4 q -6 -3 q -7 + 2 q -8 - q -9 + q -10$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$z 4 a 8 + 4 z 2 a 8 + a 8 z -2 + 4 a 8 - z 6 a 6 -6 z 4 a 6 -13 z 2 a 6 -2 a 6 z -2 -11 a 6 + 2 z 4 a 4 + 8 z 2 a 4 + a 4 z -2 + 7 a 4$ (db)
Kauffman polynomial	$z 4 a 12 -3 z 2 a 12 + a 12 + z 5 a 11 -2 z 3 a 11 + z 6 a 10 -2 z 4 a 10 + z 2 a 10 + z 7 a 9 -3 z 5 a 9 + 4 z 3 a 9 + z 8 a 8 -5 z 6 a 8 + 11 z 4 a 8 -8 z 2 a 8 - a 8 z -2 + 5 a 8 + 2 z 7 a 7 -9 z 5 a 7 + 17 z 3 a 7 -11 z a 7 + 2 a 7 z -1 + z 8 a 6 -6 z 6 a 6 + 17 z 4 a 6 -23 z 2 a 6 -2 a 6 z -2 + 13 a 6 + z 7 a 5 -5 z 5 a 5 + 11 z 3 a 5 -11 z a 5 + 2 a 5 z -1 + 3 z 4 a 4 -11 z 2 a 4 - a 4 z -2 + 8 a 4$ (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 =

-4 is the signature of L10n78. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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