L11a56

(Knotscape image)

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

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_14,8,15,7 X_16,10,17,9 X_20,12,21,11 X_22,18,5,17 X_18,22,19,21 X_8,16,9,15 X_10,20,11,19 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -8, 4, -9, 5, -2, 11, -3, 8, -4, 6, -7, 9, -5, 7, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 5 + 2 u 5 + 4 v u 4 -4 u 4 -5 v u 3 + 5 u 3 + 5 v u 2 -5 u 2 -4 v u + 4 u + 2 v -2$ (db)
Jones polynomial	$q^{23/2}-3 q^{21/2}+6 q^{19/2}-9 q^{17/2}+13 q^{15/2}-14 q^{13/2}+13 q^{11/2}-12 q^{9/2}+8 q^{7/2}-6 q^{5/2}+2 q^{3/2}-\sqrt{q}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$- z 7 a -5 - z 7 a -7 + z 5 a -3 -4 z 5 a -5 -4 z 5 a -7 + z 5 a -9 + 4 z 3 a -3 -5 z 3 a -5 -5 z 3 a -7 + 3 z 3 a -9 + 5 z a -3 -4 z a -5 -3 z a -7 + 2 z a -9 + 2 a -3 z -1 -2 a -5 z -1 - a -7 z -1 + a -9 z -1$ (db)
Kauffman polynomial	$- z 10 a -6 - z 10 a -8 -2 z 9 a -5 -6 z 9 a -7 -4 z 9 a -9 -2 z 8 a -4 -3 z 8 a -6 -7 z 8 a -8 -6 z 8 a -10 - z 7 a -3 + 3 z 7 a -5 + 15 z 7 a -7 + 5 z 7 a -9 -6 z 7 a -11 + 7 z 6 a -4 + 17 z 6 a -6 + 25 z 6 a -8 + 10 z 6 a -10 -5 z 6 a -12 + 5 z 5 a -3 + 9 z 5 a -5 -6 z 5 a -7 + z 5 a -9 + 8 z 5 a -11 -3 z 5 a -13 -5 z 4 a -4 -15 z 4 a -6 -23 z 4 a -8 -6 z 4 a -10 + 6 z 4 a -12 - z 4 a -14 -9 z 3 a -3 -16 z 3 a -5 -2 z 3 a -7 -3 z 3 a -9 -5 z 3 a -11 + 3 z 3 a -13 -3 z 2 a -4 + 4 z 2 a -6 + 10 z 2 a -8 - z 2 a -10 -3 z 2 a -12 + z 2 a -14 + 7 z a -3 + 8 z a -5 + z a -11 + 3 a -4 -3 a -8 + a -12 -2 a -3 z -1 -2 a -5 z -1 + a -7 z -1 + a -9 z -1$ (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 L11a56. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 4$	$i = 6$
$r = -2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 0$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2$	${\mathbb Z}^{3}$
$r = 1$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 4$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 5$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 6$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 7$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 8$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 9$	${\mathbb Z}_2$	${\mathbb Z}$