L11a48

(Knotscape image)

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

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Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_20,16,21,15 X_16,20,17,19 X_4,21,1,22 X_12,6,13,5 X_10,4,11,3 X_22,12,5,11 X_14,9,15,10 X_2,14,3,13 X_8,17,9,18
Gauss code	{1, -10, 7, -5}, {6, -1, 2, -11, 9, -7, 8, -6, 10, -9, 3, -4, 11, -2, 4, -3, 5, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 6 v u 4 -6 u 4 -11 v u 3 + 11 u 3 + 11 v u 2 -11 u 2 -6 v u + 6 u + v -1$ (db)
Jones polynomial	$-q^{13/2}+4 q^{11/2}-10 q^{9/2}+15 q^{7/2}-20 q^{5/2}+24 q^{3/2}-23 \sqrt{q}+\frac{19}{\sqrt{q}}-\frac{15}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 2 a z 5 -3 z 5 a -1 + 2 z 5 a -3 - a 3 z 3 + 4 a z 3 -4 z 3 a -1 + 4 z 3 a -3 - z 3 a -5 - a 3 z + a z -2 z a -1 + 3 z a -3 - z a -5 + a 3 z -1 - a z -1 - a -1 z -1 + 2 a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -2 -2 z 10 -5 a z 9 -12 z 9 a -1 -7 z 9 a -3 -6 a 2 z 8 -16 z 8 a -2 -11 z 8 a -4 -11 z 8 -4 a 3 z 7 + a z 7 + 14 z 7 a -1 -9 z 7 a -5 - a 4 z 6 + 11 a 2 z 6 + 38 z 6 a -2 + 19 z 6 a -4 -4 z 6 a -6 + 27 z 6 + 10 a 3 z 5 + 16 a z 5 + 9 z 5 a -1 + 19 z 5 a -3 + 15 z 5 a -5 - z 5 a -7 + 2 a 4 z 4 -3 a 2 z 4 -26 z 4 a -2 -14 z 4 a -4 + 4 z 4 a -6 -13 z 4 -8 a 3 z 3 -14 a z 3 -18 z 3 a -1 -23 z 3 a -3 -10 z 3 a -5 + z 3 a -7 - a 4 z 2 - a 2 z 2 + 6 z 2 a -2 + 4 z 2 a -4 + 2 z 2 + a 3 z + 3 a z + 10 z a -1 + 13 z a -3 + 5 z a -5 - a 2 -3 a -2 - a -4 -2 + a 3 z -1 + a z -1 - a -1 z -1 -2 a -3 z -1 - a -5 z -1$ (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 L11a48. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 0$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{11}$
$r = 1$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$