L11n37

(Knotscape image)

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

Visit L11n37's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_20,7,21,8 X_4,21,1,22 X_11,14,12,15 X₈₄₉₃ X_5,13,6,12 X_13,5,14,22 X_15,19,16,18 X_9,17,10,16 X_17,11,18,10 X_2,20,3,19
Gauss code	{1, -11, 5, -3}, {-6, -1, 2, -5, -9, 10, -4, 6, -7, 4, -8, 9, -10, 8, 11, -2, 3, 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	$q^{15/2}-3 q^{13/2}+6 q^{11/2}-8 q^{9/2}+9 q^{7/2}-10 q^{5/2}+8 q^{3/2}-7 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{1}{q^{3/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$2 z 5 a -3 -3 z 3 a -1 + 7 z 3 a -3 -3 z 3 a -5 + a z -5 z a -1 + 9 z a -3 -6 z a -5 + z a -7 + a z -1 -2 a -1 z -1 + 3 a -3 z -1 -3 a -5 z -1 + a -7 z -1$ (db)
Kauffman polynomial	$- z 9 a -3 - z 9 a -5 -3 z 8 a -2 -6 z 8 a -4 -3 z 8 a -6 -2 z 7 a -1 -5 z 7 a -3 -6 z 7 a -5 -3 z 7 a -7 + 9 z 6 a -2 + 15 z 6 a -4 + 5 z 6 a -6 - z 6 a -8 + 6 z 5 a -1 + 24 z 5 a -3 + 27 z 5 a -5 + 9 z 5 a -7 -15 z 4 a -2 -9 z 4 a -4 + 6 z 4 a -6 + 3 z 4 a -8 -3 z 4 - a z 3 -16 z 3 a -1 -36 z 3 a -3 -28 z 3 a -5 -7 z 3 a -7 + 7 z 2 a -2 + z 2 a -4 -7 z 2 a -6 -3 z 2 a -8 + 2 z 2 + 2 a z + 11 z a -1 + 20 z a -3 + 14 z a -5 + 3 z a -7 -2 a -2 + 2 a -6 + a -8 - a z -1 -2 a -1 z -1 -3 a -3 z -1 -3 a -5 z -1 - a -7 z -1$ (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 L11n37. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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