L10n30

(Knotscape image)

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

Visit L10n30's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_3,10,4,11 X_15,5,16,20 X_7,17,8,16 X_17,12,18,13 X_9,14,10,15 X_13,18,14,19 X_19,9,20,8 X₂₅₃₆ X_11,4,12,1
Gauss code	{1, -9, -2, 10}, {9, -1, -4, 8, -6, 2, -10, 5, -7, 6, -3, 4, -5, 7, -8, 3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 3 v u 4 - u 4 -4 v u 3 + 2 u 3 + 2 v u 2 -4 u 2 - v u + 3 u -1$ (db)
Jones polynomial	$q^{3/2}-3 \sqrt{q}+\frac{5}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{2}{q^{13/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z a 7 - a 7 z -1 + z 5 a 5 + 4 z 3 a 5 + 6 z a 5 + 4 a 5 z -1 - z 7 a 3 -5 z 5 a 3 -9 z 3 a 3 -9 z a 3 -4 a 3 z -1 + z 5 a + 3 z 3 a + 2 z a + a z -1$ (db)
Kauffman polynomial	$-3 z 2 a 8 + a 8 - z 5 a 7 -5 z 3 a 7 + 4 z a 7 - a 7 z -1 -5 z 6 a 6 + 11 z 4 a 6 -13 z 2 a 6 + 4 a 6 -6 z 7 a 5 + 18 z 5 a 5 -24 z 3 a 5 + 16 z a 5 -4 a 5 z -1 -2 z 8 a 4 -3 z 6 a 4 + 21 z 4 a 4 -21 z 2 a 4 + 7 a 4 -9 z 7 a 3 + 29 z 5 a 3 -28 z 3 a 3 + 15 z a 3 -4 a 3 z -1 -2 z 8 a 2 + z 6 a 2 + 13 z 4 a 2 -14 z 2 a 2 + 4 a 2 -3 z 7 a + 10 z 5 a -9 z 3 a + 3 z a - a z -1 - z 6 + 3 z 4 -3 z 2 + 1$ (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 =

-3 is the signature of L10n30. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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