L11n57

(Knotscape image)

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

Visit L11n57's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_7,14,8,15 X_11,19,12,18 X_19,5,20,22 X_15,21,16,20 X_21,17,22,16 X_17,13,18,12 X_13,8,14,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, -4, 8, -9, 3, -6, 7, -8, 4, -5, 6, -7, 5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$u 5 - v u 4 -3 u 4 + 2 u 3 + 2 v u 2 -3 v u - u + v$ (db)
Jones polynomial	$q^{9/2}-q^{7/2}+q^{3/2}-3 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z a 5 + a 5 z -1 -2 z 3 a 3 -4 z a 3 -2 a 3 z -1 + z 5 a + 4 z 3 a + 6 z a + 3 a z -1 - z 5 a -1 -6 z 3 a -1 -8 z a -1 -3 a -1 z -1 + z 3 a -3 + 3 z a -3 + a -3 z -1$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 -2 a 4 z 8 -4 a 2 z 8 -2 z 8 - a 5 z 7 + a 3 z 7 + 2 a z 7 - z 7 a -1 - z 7 a -3 + 9 a 4 z 6 + 19 a 2 z 6 - z 6 a -4 + 11 z 6 + 5 a 5 z 5 + 12 a 3 z 5 + 10 a z 5 + 9 z 5 a -1 + 6 z 5 a -3 -10 a 4 z 4 -23 a 2 z 4 + 4 z 4 a -2 + 5 z 4 a -4 -14 z 4 -8 a 5 z 3 -20 a 3 z 3 -22 a z 3 -18 z 3 a -1 -8 z 3 a -3 + 3 a 4 z 2 + 10 a 2 z 2 -7 z 2 a -2 -5 z 2 a -4 + 5 z 2 + 5 a 5 z + 10 a 3 z + 14 a z + 13 z a -1 + 4 z a -3 -2 a 2 + 2 a -2 + a -4 - a 5 z -1 -2 a 3 z -1 -3 a z -1 -3 a -1 z -1 - a -3 z -1$ (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 L11n57. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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