L11n54

(Knotscape image)

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

Visit L11n54's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 u 5 + 2 u 4 - v u 3 - u 2 + 2 v u -2 v$ (db)
Jones polynomial	$q^{15/2}-q^{13/2}+q^{9/2}-q^{7/2}+2 q^{5/2}-3 q^{3/2}+2 \sqrt{q}-\frac{3}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$- z 5 a -1 - z 5 a -3 + a z 3 -4 z 3 a -1 -4 z 3 a -3 + 3 a z -4 z a -1 -3 z a -3 + z a -5 + z a -7 + 2 a z -1 -2 a -1 z -1 - a -3 z -1 + a -5 z -1$ (db)
Kauffman polynomial	$- z 9 a -1 - z 9 a -3 -3 z 8 a -2 -2 z 8 a -4 - z 8 - a z 7 + 4 z 7 a -1 + 5 z 7 a -3 - z 7 a -5 - z 7 a -7 + 16 z 6 a -2 + 13 z 6 a -4 - z 6 a -8 + 4 z 6 + 6 a z 5 - z 5 a -1 -6 z 5 a -3 + 7 z 5 a -5 + 6 z 5 a -7 -22 z 4 a -2 -24 z 4 a -4 + 2 z 4 a -6 + 5 z 4 a -8 - z 4 -11 a z 3 -5 z 3 a -1 + 5 z 3 a -3 -9 z 3 a -5 -8 z 3 a -7 + 9 z 2 a -2 + 17 z 2 a -4 -2 z 2 a -6 -5 z 2 a -8 -5 z 2 + 8 a z + 5 z a -1 -4 z a -3 + z a -5 + 2 z a -7 -3 a -4 + a -8 + 3-2 a z -1 -2 a -1 z -1 + a -3 z -1 + a -5 z -1$ (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 =

3 is the signature of L11n54. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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