L11n72

(Knotscape image)

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

Visit L11n72's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_7,16,8,17 X_22,18,5,17 X_18,12,19,11 X_12,22,13,21 X_20,14,21,13 X_14,20,15,19 X_15,8,16,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, 5, -6, 7, -8, -9, 3, 4, -5, 8, -7, 6, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 2 u 3 + 5 v u 2 -6 u 2 -6 v u + 5 u + 2 v -2$ (db)
Jones polynomial	$q^{9/2}-3 q^{7/2}+5 q^{5/2}-8 q^{3/2}+10 \sqrt{q}-\frac{10}{\sqrt{q}}+\frac{9}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{2}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 5 z -1 + z 3 a 3 - z a 3 - a 3 z -1 - z 5 a - z 3 a - z 5 a -1 -2 z 3 a -1 -2 z a -1 + z 3 a -3 + z a -3$ (db)
Kauffman polynomial	$- a z 9 - z 9 a -1 -2 a 2 z 8 -3 z 8 a -2 -5 z 8 -2 a 3 z 7 -3 a z 7 -4 z 7 a -1 -3 z 7 a -3 - a 4 z 6 - a 2 z 6 + 7 z 6 a -2 - z 6 a -4 + 8 z 6 - a 3 z 5 + 4 a z 5 + 15 z 5 a -1 + 10 z 5 a -3 -2 a 4 z 4 + 2 a 2 z 4 -2 z 4 a -2 + 3 z 4 a -4 - z 4 -3 a 5 z 3 + 3 a z 3 -9 z 3 a -1 -9 z 3 a -3 - a 4 z 2 - z 2 a -2 -2 z 2 a -4 + 2 z 2 + 3 a 5 z + a 3 z -2 a z + z a -1 + z a -3 + a 4 - a 5 z -1 - a 3 z -1$ (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 L11n72. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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