L11a51

(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_18,9,19,10 X_8,17,9,18 X_4,19,1,20 X_14,6,15,5 X_10,4,11,3 X_20,12,21,11 X_22,14,5,13 X_12,22,13,21 X_2,16,3,15
Gauss code	{1, -11, 7, -5}, {6, -1, 2, -4, 3, -7, 8, -10, 9, -6, 11, -2, 4, -3, 5, -8, 10, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 7 + u 7 + 2 v u 6 -2 u 6 -3 v u 5 + 3 u 5 + 4 v u 4 -4 u 4 -4 v u 3 + 4 u 3 + 3 v u 2 -3 u 2 -2 v u + 2 u + v -1$ (db)
Jones polynomial	$-q^{13/2}+3 q^{11/2}-5 q^{9/2}+8 q^{7/2}-11 q^{5/2}+12 q^{3/2}-13 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z 9 a -1 - a z 7 + 7 z 7 a -1 - z 7 a -3 -5 a z 5 + 17 z 5 a -1 -5 z 5 a -3 -7 a z 3 + 16 z 3 a -1 -7 z 3 a -3 -2 a z + 4 z a -1 -2 z a -3 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -2 -2 z 10 -4 a z 9 -8 z 9 a -1 -4 z 9 a -3 -4 a 2 z 8 + 4 z 8 a -2 -4 z 8 a -4 + 4 z 8 -3 a 3 z 7 + 15 a z 7 + 36 z 7 a -1 + 14 z 7 a -3 -4 z 7 a -5 - a 4 z 6 + 13 a 2 z 6 + 9 z 6 a -4 -3 z 6 a -6 + 2 z 6 + 10 a 3 z 5 -21 a z 5 -65 z 5 a -1 -24 z 5 a -3 + 9 z 5 a -5 - z 5 a -7 + 3 a 4 z 4 -9 a 2 z 4 -11 z 4 a -2 -6 z 4 a -4 + 7 z 4 a -6 -10 z 4 -6 a 3 z 3 + 16 a z 3 + 44 z 3 a -1 + 16 z 3 a -3 -4 z 3 a -5 + 2 z 3 a -7 - a 4 z 2 + a 2 z 2 + 5 z 2 a -2 -2 z 2 a -6 + 5 z 2 -4 a z -8 z a -1 -4 z a -3 + 1- a z -1 - a -1 z -1$ (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 L11a51. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -2$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -1$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 0$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{7}$
$r = 1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 2$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 4$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$