L10n34

(Knotscape image)

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

Visit L10n34's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_16,8,17,7 X_17,20,18,5 X_11,19,12,18 X_19,11,20,10 X_14,10,15,9 X_8,16,9,15 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -8, 7, 6, -5, -2, 10, -7, 8, -3, -4, 5, -6, 4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-3 v u + 3 u + 3 v -3$ (db)
Jones polynomial	$q^{15/2}-2 q^{13/2}+3 q^{11/2}-4 q^{9/2}+4 q^{7/2}-4 q^{5/2}+3 q^{3/2}-3 \sqrt{q}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-2 z 3 a -3 - z 3 a -5 + 3 z a -1 -4 z a -3 + z a -7 + 2 a -1 z -1 -3 a -3 z -1 + a -5 z -1$ (db)
Kauffman polynomial	$- z 8 a -4 - z 8 a -6 - z 7 a -3 -3 z 7 a -5 -2 z 7 a -7 + 3 z 6 a -4 + 2 z 6 a -6 - z 6 a -8 + z 5 a -3 + 9 z 5 a -5 + 8 z 5 a -7 -3 z 4 a -2 -6 z 4 a -4 + z 4 a -6 + 4 z 4 a -8 + 3 z 3 a -3 -5 z 3 a -5 -8 z 3 a -7 + 4 z 2 a -2 + 9 z 2 a -4 + z 2 a -6 -4 z 2 a -8 -5 z a -1 -7 z a -3 + 2 z a -7 -3 a -2 -3 a -4 - a -6 + 2 a -1 z -1 + 3 a -3 z -1 + a -5 z -1$ (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 L10n34. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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