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 $u$ , $v$ , $w$ , ...)	$\frac{u v^3 w^2+u v^2 w^3-u v^2 w^2+v w-v-w}{\sqrt{u} v^{3/2} w^{3/2}}$ (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	$a 14 z 2 -2 a 14 + a 13 z + a 12 z 4 -3 a 12 z 2 - a 12 z -2 + 3 a 12 + a 11 z 7 -6 a 11 z 5 + 12 a 11 z 3 -8 a 11 z + 2 a 11 z -1 + a 10 z 8 -6 a 10 z 6 + 13 a 10 z 4 -15 a 10 z 2 -2 a 10 z -2 + 9 a 10 + 2 a 9 z 7 -10 a 9 z 5 + 14 a 9 z 3 -8 a 9 z + 2 a 9 z -1 + a 8 z 8 -5 a 8 z 6 + 7 a 8 z 4 -5 a 8 z 2 - a 8 z -2 + 3 a 8 + a 7 z 7 -4 a 7 z 5 + 2 a 7 z 3 + a 7 z + a 6 z 6 -5 a 6 z 4 + 6 a 6 z 2 -2 a 6$ (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.

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