L10n74

(Knotscape image)

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

Visit L10n74's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_5,14,6,15 X₃₈₄₉ X_15,2,16,3 X_16,7,17,8 X_9,18,10,19 X_4,17,1,18 X_19,12,20,5 X_11,20,12,13 X_13,10,14,11
Gauss code	{1, 4, -3, -7}, {-2, -1, 5, 3, -6, 10, -9, 8}, {-10, 2, -4, -5, 7, 6, -8, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 w u 3 - v 3 w u 2 + v 2 w u 2 - v u + u + v$ (db)
Jones polynomial	$q -3 - q -4 + 2 q -5 - q -6 + 2 q -7 -2 q -8 + 2 q -9 + q -11$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$a 12 z -2 + a 12 - z 4 a 10 -5 z 2 a 10 -2 a 10 z -2 -5 a 10 + z 6 a 8 + 5 z 4 a 8 + 6 z 2 a 8 + a 8 z -2 + 2 a 8 + z 6 a 6 + 5 z 4 a 6 + 6 z 2 a 6 + 2 a 6$ (db)
Kauffman polynomial	$z 2 a 14 -2 a 14 + z a 13 + z 4 a 12 -3 z 2 a 12 - a 12 z -2 + 3 a 12 + z 7 a 11 -6 z 5 a 11 + 12 z 3 a 11 -8 z a 11 + 2 a 11 z -1 + z 8 a 10 -6 z 6 a 10 + 13 z 4 a 10 -15 z 2 a 10 -2 a 10 z -2 + 9 a 10 + 2 z 7 a 9 -10 z 5 a 9 + 14 z 3 a 9 -8 z a 9 + 2 a 9 z -1 + z 8 a 8 -5 z 6 a 8 + 7 z 4 a 8 -5 z 2 a 8 - a 8 z -2 + 3 a 8 + z 7 a 7 -4 z 5 a 7 + 2 z 3 a 7 + z a 7 + z 6 a 6 -5 z 4 a 6 + 6 z 2 a 6 -2 a 6$ (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 =

-6 is the signature of L10n74. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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