L10n38

(Knotscape image)

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

Visit L10n38's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_16,13,17,14 X_14,9,15,10 X_10,15,11,16 X_17,5,18,20 X_7,19,8,18 X_19,9,20,8 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -9, 2, -10}, {9, -1, -7, 8, 4, -5, 10, -2, 3, -4, 5, -3, -6, 7, -8, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 5 - v u 4 + 2 u 4 + 2 v u 3 -3 u 3 -3 v u 2 + 2 u 2 + 2 v u - u - v$ (db)
Jones polynomial	$2 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{5}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{2}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 -2 a 7 z -1 + 3 z 3 a 5 + 9 z a 5 + 7 a 5 z -1 -2 z 5 a 3 -9 z 3 a 3 -14 z a 3 -7 a 3 z -1 + 2 z 3 a + 4 z a + 2 a z -1$ (db)
Kauffman polynomial	$- z 6 a 8 + 4 z 4 a 8 -5 z 2 a 8 + 2 a 8 -2 z 7 a 7 + 7 z 5 a 7 -7 z 3 a 7 + 4 z a 7 -2 a 7 z -1 - z 8 a 6 -2 z 6 a 6 + 17 z 4 a 6 -20 z 2 a 6 + 8 a 6 -6 z 7 a 5 + 20 z 5 a 5 -22 z 3 a 5 + 17 z a 5 -7 a 5 z -1 - z 8 a 4 -5 z 6 a 4 + 24 z 4 a 4 -29 z 2 a 4 + 13 a 4 -4 z 7 a 3 + 12 z 5 a 3 -18 z 3 a 3 + 16 z a 3 -7 a 3 z -1 -4 z 6 a 2 + 11 z 4 a 2 -17 z 2 a 2 + 8 a 2 - z 5 a -3 z 3 a + 3 z a -2 a z -1 -3 z 2 + 2$ (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 L10n38. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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