L10n17

(Knotscape image)

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

Visit L10n17's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_4,19,1,20 X_5,12,6,13 X₃₈₄₉ X_13,17,14,16 X_9,15,10,14 X_15,11,16,10 X_11,20,12,5 X_17,2,18,3
Gauss code	{1, 10, -5, -3}, {-4, -1, 2, 5, -7, 8, -9, 4, -6, 7, -8, 6, -10, -2, 3, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 2 v u 4 -3 v u 3 + u 3 + v u 2 -3 u 2 + 2 u -1$ (db)
Jones polynomial	$q^{3/2}-2 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{1}{q^{13/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z a 7 - a 7 z -1 + z 5 a 5 + 5 z 3 a 5 + 8 z a 5 + 4 a 5 z -1 - z 7 a 3 -6 z 5 a 3 -13 z 3 a 3 -13 z a 3 -4 a 3 z -1 + z 5 a + 4 z 3 a + 4 z a + a z -1$ (db)
Kauffman polynomial	$- z 2 a 8 + a 8 -3 z 3 a 7 + 2 z a 7 - a 7 z -1 -2 z 6 a 6 + 6 z 4 a 6 -10 z 2 a 6 + 4 a 6 -3 z 7 a 5 + 12 z 5 a 5 -20 z 3 a 5 + 14 z a 5 -4 a 5 z -1 - z 8 a 4 - z 6 a 4 + 13 z 4 a 4 -18 z 2 a 4 + 7 a 4 -5 z 7 a 3 + 20 z 5 a 3 -26 z 3 a 3 + 17 z a 3 -4 a 3 z -1 - z 8 a 2 + 11 z 4 a 2 -13 z 2 a 2 + 4 a 2 -2 z 7 a + 8 z 5 a -9 z 3 a + 5 z a - a z -1 - z 6 + 4 z 4 -4 z 2 + 1$ (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 L10n17. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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