L11n84

(Knotscape image)

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

Visit L11n84's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_14,8,15,7 X_9,18,10,19 X_19,5,20,22 X_15,21,16,20 X_21,17,22,16 X_17,8,18,9 X_10,14,11,13 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, 8, -4, -9, 11, -2, 9, -3, -6, 7, -8, 4, -5, 6, -7, 5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + u 3 + 3 v u 2 -4 u 2 -4 v u + 3 u + v -1$ (db)
Jones polynomial	$q^{9/2}-3 q^{7/2}+4 q^{5/2}-5 q^{3/2}+6 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{5}{q^{3/2}}-\frac{4}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 5 z -1 -2 z a 3 - a 3 z -1 + 2 z 3 a + 2 z a - z 5 a -1 -3 z 3 a -1 -3 z a -1 + z 3 a -3 + z a -3$ (db)
Kauffman polynomial	$- a z 9 - z 9 a -1 - a 2 z 8 -3 z 8 a -2 -4 z 8 + 3 a z 7 -3 z 7 a -3 + 4 a 2 z 6 + 11 z 6 a -2 - z 6 a -4 + 16 z 6 -2 a 3 z 5 -3 a z 5 + 10 z 5 a -1 + 11 z 5 a -3 - a 4 z 4 -9 a 2 z 4 -9 z 4 a -2 + 3 z 4 a -4 -20 z 4 - a 5 z 3 + 2 a 3 z 3 -11 z 3 a -1 -8 z 3 a -3 + 5 a 2 z 2 + 2 z 2 a -2 - z 2 a -4 + 8 z 2 + 2 a 5 z - a z + 2 z a -1 + z a -3 + a 4 - a 5 z -1 - a 3 z -1$ (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 L11n84. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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