L11a57

(Knotscape image)

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

Visit L11a57's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 5 v u 4 -5 u 4 -9 v u 3 + 9 u 3 + 9 v u 2 -9 u 2 -5 v u + 5 u + v -1$ (db)
Jones polynomial	$q^{11/2}-4 q^{9/2}+8 q^{7/2}-13 q^{5/2}+17 q^{3/2}-20 \sqrt{q}+\frac{18}{\sqrt{q}}-\frac{16}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 3 a z 5 -4 z 5 a -1 + z 5 a -3 -3 a 3 z 3 + 9 a z 3 -7 z 3 a -1 + 2 z 3 a -3 + a 5 z -6 a 3 z + 9 a z -5 z a -1 + z a -3 + a 5 z -1 -3 a 3 z -1 + 4 a z -1 -2 a -1 z -1$ (db)
Kauffman polynomial	$- a 2 z 10 - z 10 -3 a 3 z 9 -8 a z 9 -5 z 9 a -1 -3 a 4 z 8 -11 a 2 z 8 -10 z 8 a -2 -18 z 8 - a 5 z 7 + 3 a 3 z 7 + 5 a z 7 -10 z 7 a -1 -11 z 7 a -3 + 11 a 4 z 6 + 44 a 2 z 6 + 10 z 6 a -2 -8 z 6 a -4 + 51 z 6 + 4 a 5 z 5 + 17 a 3 z 5 + 42 a z 5 + 47 z 5 a -1 + 14 z 5 a -3 -4 z 5 a -5 -14 a 4 z 4 -47 a 2 z 4 + 3 z 4 a -2 + 7 z 4 a -4 - z 4 a -6 -38 z 4 -6 a 5 z 3 -32 a 3 z 3 -60 a z 3 -44 z 3 a -1 -8 z 3 a -3 + 2 z 3 a -5 + 7 a 4 z 2 + 19 a 2 z 2 -2 z 2 a -2 -2 z 2 a -4 + 12 z 2 + 4 a 5 z + 18 a 3 z + 26 a z + 15 z a -1 + 3 z a -3 - a 4 -3 a 2 - a -2 -2- a 5 z -1 -3 a 3 z -1 -4 a z -1 -2 a -1 z -1$ (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 L11a57. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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