L11n3

(Knotscape image)

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

Visit L11n3's page at the original Knot Atlas.

Link L11n3.

A graph, L11n3.

A part of a knot and a part of a graph.

Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_5,10,6,11 X₈₄₉₃ X_11,20,12,21 X_17,5,18,22 X_21,19,22,18 X_19,12,20,13 X_9,16,10,17 X_2,14,3,13
Gauss code	{1, -11, 5, -3}, {-4, -1, 2, -5, -10, 4, -6, 9, 11, -2, 3, 10, -7, 8, -9, 6, -8, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + u 3 + 4 v u 2 -4 u 2 -4 v u + 4 u + v -1$ (db)
Jones polynomial	$-q^{3/2}+2 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{6}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{2}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 - a 7 z -1 + 2 z 3 a 5 + 4 z a 5 + 3 a 5 z -1 - z 5 a 3 -3 z 3 a 3 -5 z a 3 -3 a 3 z -1 + 2 z 3 a + 3 z a + 2 a z -1 - z a -1 - a -1 z -1$ (db)
Kauffman polynomial	$- a 5 z 9 - a 3 z 9 -2 a 6 z 8 -4 a 4 z 8 -2 a 2 z 8 -2 a 7 z 7 + a 3 z 7 - a z 7 - a 8 z 6 + 5 a 6 z 6 + 14 a 4 z 6 + 8 a 2 z 6 + 7 a 7 z 5 + 9 a 5 z 5 + 5 a 3 z 5 + 3 a z 5 + 4 a 8 z 4 + a 6 z 4 -16 a 4 z 4 -15 a 2 z 4 -2 z 4 -6 a 7 z 3 -13 a 5 z 3 -15 a 3 z 3 -9 a z 3 - z 3 a -1 -4 a 8 z 2 -4 a 6 z 2 + 6 a 4 z 2 + 8 a 2 z 2 + 2 z 2 + 3 a 7 z + 10 a 5 z + 12 a 3 z + 7 a z + 2 z a -1 + a 8 + 2 a 6 -2 a 2 - a 7 z -1 -3 a 5 z -1 -3 a 3 z -1 -2 a z -1 - a -1 z -1$ (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 L11n3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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