L11n63

(Knotscape image)

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

Visit L11n63's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_7,14,8,15 X_20,16,21,15 X_18,12,19,11 X_12,20,13,19 X_22,18,5,17 X_16,22,17,21 X_13,8,14,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, 5, -6, -9, 3, 4, -8, 7, -5, 6, -4, 8, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 2 v u 4 -4 u 4 -4 v u 3 + 5 u 3 + 5 v u 2 -4 u 2 -4 v u + 2 u + v -1$ (db)
Jones polynomial	$q^{11/2}-3 q^{9/2}+7 q^{7/2}-10 q^{5/2}+11 q^{3/2}-12 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{2}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + a z 5 -5 z 5 a -1 + z 5 a -3 + 2 a z 3 -10 z 3 a -1 + 3 z 3 a -3 + a 3 z + a z -8 z a -1 + 4 z a -3 + a 3 z -1 -2 a -1 z -1 + a -3 z -1$ (db)
Kauffman polynomial	$- a z 9 - z 9 a -1 - a 2 z 8 -4 z 8 a -2 -5 z 8 - a z 7 -8 z 7 a -1 -7 z 7 a -3 - z 6 a -2 -6 z 6 a -4 + 5 z 6 -3 a 3 z 5 + 2 a z 5 + 19 z 5 a -1 + 11 z 5 a -3 -3 z 5 a -5 + 2 a 2 z 4 + 14 z 4 a -2 + 8 z 4 a -4 - z 4 a -6 + 7 z 4 + 8 a 3 z 3 + 4 a z 3 -15 z 3 a -1 -9 z 3 a -3 + 2 z 3 a -5 + a 2 z 2 -16 z 2 a -2 -6 z 2 a -4 + z 2 a -6 -8 z 2 -6 a 3 z -2 a z + 8 z a -1 + 4 z a -3 - a 2 + 5 a -2 + 2 a -4 + 3 + a 3 z -1 -2 a -1 z -1 - a -3 z -1$ (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 L11n63. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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