L11n31

(Knotscape image)

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

Visit L11n31'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 X₈₄₉₃ X_9,16,10,17 X_13,22,14,5 X_15,10,16,11 X_21,14,22,15 X_11,20,12,21 X_2,18,3,17
Gauss code	{1, -11, 5, -3}, {-4, -1, 2, -5, -6, 8, -10, 4, -7, 9, -8, 6, 11, -2, 3, 10, -9, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 2 u 3 + 5 v u 2 -5 u 2 -5 v u + 5 u + 2 v -2$ (db)
Jones polynomial	$-\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{10}{q^{7/2}}+\frac{9}{q^{9/2}}-\frac{9}{q^{11/2}}+\frac{6}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{2}{q^{17/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a 9 z -1 - z 3 a 7 + a 7 z -1 + z 5 a 5 + 2 z 3 a 5 + 3 z a 5 + 2 a 5 z -1 + z 5 a 3 + z 3 a 3 -2 z a 3 -2 a 3 z -1 - z 3 a - z a$ (db)
Kauffman polynomial	$-3 z 4 a 10 + 8 z 2 a 10 -4 a 10 - z 7 a 9 + z 5 a 9 - z 3 a 9 + 2 z a 9 + a 9 z -1 -2 z 8 a 8 + 7 z 6 a 8 -18 z 4 a 8 + 21 z 2 a 8 -9 a 8 - z 9 a 7 - z 7 a 7 + 8 z 5 a 7 -17 z 3 a 7 + 7 z a 7 + a 7 z -1 -5 z 8 a 6 + 14 z 6 a 6 -20 z 4 a 6 + 11 z 2 a 6 -4 a 6 - z 9 a 5 -4 z 7 a 5 + 16 z 5 a 5 -22 z 3 a 5 + 11 z a 5 -2 a 5 z -1 -3 z 8 a 4 + 4 z 6 a 4 + z 4 a 4 -3 z 2 a 4 + 2 a 4 -4 z 7 a 3 + 8 z 5 a 3 -4 z 3 a 3 + 5 z a 3 -2 a 3 z -1 -3 z 6 a 2 + 6 z 4 a 2 - z 2 a 2 - z 5 a + 2 z 3 a - z a$ (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 L11n31. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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