L11n50

(Knotscape image)

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

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Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,8,13,7 X_13,18,14,19 X_9,17,10,16 X_17,9,18,8 X_19,22,20,5 X_15,21,16,20 X_21,15,22,14 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, 6, -5, -2, 11, -3, -4, 9, -8, 5, -6, 4, -7, 8, -9, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + u 3 + 5 v u 2 -5 u 2 -5 v u + 5 u + v -1$ (db)
Jones polynomial	$q^{15/2}-3 q^{13/2}+5 q^{11/2}-7 q^{9/2}+8 q^{7/2}-8 q^{5/2}+7 q^{3/2}-6 \sqrt{q}+\frac{2}{\sqrt{q}}-\frac{1}{q^{3/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z 5 a -3 -2 z 3 a -1 + 2 z 3 a -3 -2 z 3 a -5 + a z -2 z a -1 + 2 z a -3 -2 z a -5 + z a -7 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$- z 9 a -3 - z 9 a -5 -2 z 8 a -2 -5 z 8 a -4 -3 z 8 a -6 - z 7 a -1 - z 7 a -3 -3 z 7 a -5 -3 z 7 a -7 + 6 z 6 a -2 + 15 z 6 a -4 + 8 z 6 a -6 - z 6 a -8 + 2 z 5 a -1 + 9 z 5 a -3 + 17 z 5 a -5 + 10 z 5 a -7 -10 z 4 a -2 -14 z 4 a -4 -3 z 4 a -6 + 3 z 4 a -8 -2 z 4 - a z 3 -7 z 3 a -1 -13 z 3 a -3 -15 z 3 a -5 -8 z 3 a -7 + 3 z 2 a -2 + 5 z 2 a -4 + z 2 a -6 -2 z 2 a -8 + z 2 + 2 a z + 4 z a -1 + 4 z a -3 + 4 z a -5 + 2 z a -7 + 1- a z -1 - a -1 z -1$ (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 L11n50. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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