L11a39

(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_16,8,17,7 X_20,13,21,14 X_22,18,5,17 X_18,22,19,21 X_14,19,15,20 X_12,10,13,9 X_8,16,9,15 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 8, -2, 11, -8, 4, -7, 9, -3, 5, -6, 7, -4, 6, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-3 v u 3 + 3 u 3 + 11 v u 2 -11 u 2 -11 v u + 11 u + 3 v -3$ (db)
Jones polynomial	$q^{15/2}-4 q^{13/2}+8 q^{11/2}-13 q^{9/2}+16 q^{7/2}-18 q^{5/2}+18 q^{3/2}-15 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z 5 a -1 + 2 z 5 a -3 -2 a z 3 - z 3 a -1 + 4 z 3 a -3 -3 z 3 a -5 + a 3 z -2 a z -2 z a -1 + 5 z a -3 -3 z a -5 + z a -7 + a 3 z -1 - a z -1 - a -1 z -1 + 2 a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$- z 10 a -2 - z 10 a -4 -3 z 9 a -1 -7 z 9 a -3 -4 z 9 a -5 -10 z 8 a -2 -12 z 8 a -4 -6 z 8 a -6 -4 z 8 -3 a z 7 -2 z 7 a -1 + 4 z 7 a -3 - z 7 a -5 -4 z 7 a -7 -2 a 2 z 6 + 27 z 6 a -2 + 34 z 6 a -4 + 13 z 6 a -6 - z 6 a -8 + 5 z 6 - a 3 z 5 + 2 a z 5 + 12 z 5 a -1 + 23 z 5 a -3 + 24 z 5 a -5 + 10 z 5 a -7 + 3 a 2 z 4 -29 z 4 a -2 -26 z 4 a -4 -5 z 4 a -6 + 2 z 4 a -8 -7 z 4 + 3 a 3 z 3 + 3 a z 3 -19 z 3 a -1 -36 z 3 a -3 -24 z 3 a -5 -7 z 3 a -7 + 12 z 2 a -2 + 7 z 2 a -4 + z 2 a -6 - z 2 a -8 + 7 z 2 -3 a 3 z -3 a z + 10 z a -1 + 17 z a -3 + 9 z a -5 + 2 z a -7 - a 2 -3 a -2 - a -4 -2 + a 3 z -1 + a z -1 - a -1 z -1 -2 a -3 z -1 - a -5 z -1$ (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 L11a39. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -4$	${\mathbb Z}$
$r = -3$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 0$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{7}$
$r = 1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 3$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 4$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 6$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$