L11n5

(Knotscape image)

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

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_17,1,18,4 X_9,14,10,15 X₃₈₄₉ X_5,11,6,10 X_11,19,12,18 X_13,20,14,21 X_19,5,20,22 X_21,12,22,13 X_2,16,3,15
Gauss code	{1, -11, -5, 3}, {-6, -1, 2, 5, -4, 6, -7, 10, -8, 4, 11, -2, -3, 7, -9, 8, -10, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + u 3 + 3 v u 2 -3 u 2 -3 v u + 3 u + v -1$ (db)
Jones polynomial	$q^{7/2}-3 q^{5/2}+3 q^{3/2}-5 \sqrt{q}+\frac{5}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z a 5 + a 5 z -1 -2 z 3 a 3 -5 z a 3 -3 a 3 z -1 + z 5 a + 4 z 3 a + 7 z a + 4 a z -1 -2 z 3 a -1 -4 z a -1 -2 a -1 z -1 + z a -3$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 -2 a 4 z 8 -5 a 2 z 8 -3 z 8 - a 5 z 7 + a 3 z 7 - a z 7 -3 z 7 a -1 + 10 a 4 z 6 + 24 a 2 z 6 - z 6 a -2 + 13 z 6 + 5 a 5 z 5 + 13 a 3 z 5 + 21 a z 5 + 13 z 5 a -1 -14 a 4 z 4 -33 a 2 z 4 + z 4 a -2 -18 z 4 -7 a 5 z 3 -26 a 3 z 3 -37 a z 3 -21 z 3 a -1 -3 z 3 a -3 + 7 a 4 z 2 + 17 a 2 z 2 - z 2 a -4 + 11 z 2 + 4 a 5 z + 16 a 3 z + 22 a z + 13 z a -1 + 3 z a -3 - a 4 -3 a 2 - a -2 -2- a 5 z -1 -3 a 3 z -1 -4 a z -1 -2 a -1 z -1$ (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 L11n5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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