L11n38

(Knotscape image)

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

Visit L11n38's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_20,7,21,8 X_4,21,1,22 X_5,12,6,13 X₃₈₄₉ X_9,16,10,17 X_13,19,14,18 X_17,15,18,14 X_15,10,16,11 X_11,22,12,5 X_19,2,20,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -6, 9, -10, 4, -7, 8, -9, 6, -8, 7, -11, -2, 3, 10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 4 v u 4 -3 v u 3 + u 3 + v u 2 -3 u 2 + 4 u -1$ (db)
Jones polynomial	$-\frac{1}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{6}{q^{13/2}}+\frac{4}{q^{15/2}}-\frac{2}{q^{17/2}}+\frac{1}{q^{19/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- a 11 z -1 + z 3 a 9 + 5 z a 9 + 3 a 9 z -1 -2 z 5 a 7 -9 z 3 a 7 -10 z a 7 -3 a 7 z -1 + z 7 a 5 + 5 z 5 a 5 + 8 z 3 a 5 + 7 z a 5 + 2 a 5 z -1 - z 5 a 3 -4 z 3 a 3 -4 z a 3 - a 3 z -1$ (db)
Kauffman polynomial	$- z 2 a 12 + a 12 -2 z 3 a 11 + 2 z a 11 - a 11 z -1 - z 6 a 10 + 2 z 4 a 10 -4 z 2 a 10 + 2 a 10 -3 z 7 a 9 + 12 z 5 a 9 -18 z 3 a 9 + 11 z a 9 -3 a 9 z -1 -3 z 8 a 8 + 11 z 6 a 8 -11 z 4 a 8 + 4 z 2 a 8 - z 9 a 7 -2 z 7 a 7 + 22 z 5 a 7 -32 z 3 a 7 + 16 z a 7 -3 a 7 z -1 -5 z 8 a 6 + 21 z 6 a 6 -24 z 4 a 6 + 11 z 2 a 6 -2 a 6 - z 9 a 5 + 15 z 5 a 5 -24 z 3 a 5 + 12 z a 5 -2 a 5 z -1 -2 z 8 a 4 + 9 z 6 a 4 -11 z 4 a 4 + 4 z 2 a 4 - z 7 a 3 + 5 z 5 a 3 -8 z 3 a 3 + 5 z a 3 - a 3 z -1$ (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 =

-5 is the signature of L11n38. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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