L11n52

(Knotscape image)

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

Visit L11n52's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,8,13,7 X_13,18,14,19 X_16,9,17,10 X_8,17,9,18 X_19,22,20,5 X_15,21,16,20 X_21,15,22,14 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 5, -2, 11, -3, -4, 9, -8, -5, 6, 4, -7, 8, -9, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + u 3 + 3 v u 2 -3 u 2 -3 v u + 3 u + v -1$ (db)
Jones polynomial	$-2 q^{5/2}+3 q^{3/2}-5 \sqrt{q}+\frac{5}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z a 5 + a 5 z -1 -2 z 3 a 3 -4 z a 3 -2 a 3 z -1 + z 5 a + 3 z 3 a + 3 z a + a z -1 - z 3 a -1 + a -1 z -1 - a -3 z -1$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 -2 a 4 z 8 -5 a 2 z 8 -3 z 8 - a 5 z 7 - a z 7 -2 z 7 a -1 + 9 a 4 z 6 + 22 a 2 z 6 + 13 z 6 + 5 a 5 z 5 + 16 a 3 z 5 + 19 a z 5 + 8 z 5 a -1 -11 a 4 z 4 -26 a 2 z 4 - z 4 a -2 -16 z 4 -8 a 5 z 3 -27 a 3 z 3 -28 a z 3 -9 z 3 a -1 + 5 a 4 z 2 + 12 a 2 z 2 + z 2 a -2 + 8 z 2 + 5 a 5 z + 15 a 3 z + 12 a z + z a -1 - z a -3 - a 4 -3 a 2 - a -2 -2- a 5 z -1 -2 a 3 z -1 - a z -1 + a -1 z -1 + a -3 z -1$ (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 L11n52. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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