L11a55

(Knotscape image)

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

Visit L11a55's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 5 v u 4 -5 u 4 -11 v u 3 + 11 u 3 + 11 v u 2 -11 u 2 -5 v u + 5 u + v -1$ (db)
Jones polynomial	$-q^{13/2}+4 q^{11/2}-9 q^{9/2}+14 q^{7/2}-19 q^{5/2}+22 q^{3/2}-22 \sqrt{q}+\frac{18}{\sqrt{q}}-\frac{14}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 2 a z 5 -4 z 5 a -1 + 2 z 5 a -3 - a 3 z 3 + 5 a z 3 -9 z 3 a -1 + 5 z 3 a -3 - z 3 a -5 - a 3 z + 5 a z -9 z a -1 + 6 z a -3 - z a -5 + 2 a z -1 -4 a -1 z -1 + 3 a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$- z 10 a -2 - z 10 -4 a z 9 -9 z 9 a -1 -5 z 9 a -3 -6 a 2 z 8 -20 z 8 a -2 -9 z 8 a -4 -17 z 8 -4 a 3 z 7 -5 a z 7 - z 7 a -1 -8 z 7 a -3 -8 z 7 a -5 - a 4 z 6 + 12 a 2 z 6 + 50 z 6 a -2 + 13 z 6 a -4 -4 z 6 a -6 + 46 z 6 + 10 a 3 z 5 + 34 a z 5 + 45 z 5 a -1 + 35 z 5 a -3 + 13 z 5 a -5 - z 5 a -7 + 2 a 4 z 4 -5 a 2 z 4 -45 z 4 a -2 -8 z 4 a -4 + 5 z 4 a -6 -39 z 4 -8 a 3 z 3 -37 a z 3 -58 z 3 a -1 -38 z 3 a -3 -8 z 3 a -5 + z 3 a -7 - a 4 z 2 + a 2 z 2 + 17 z 2 a -2 + 4 z 2 a -4 - z 2 a -6 + 14 z 2 + 3 a 3 z + 15 a z + 26 z a -1 + 18 z a -3 + 4 z a -5 - a 2 -3 a -2 - a -4 -2-2 a z -1 -4 a -1 z -1 -3 a -3 z -1 - a -5 z -1$ (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 L11a55. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 0$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{11}$
$r = 1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$