L11n364

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,8,13,7 X_15,17,16,22 X_9,18,10,19 X_17,8,18,9 X_13,21,14,20 X_21,15,22,14 X_19,5,20,16 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {-6, 5, -9, 7, -8, 4}, {10, -1, 3, 6, -5, -2, 11, -3, -7, 8, -4, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 2 -2 v u 2 - v 2 w u 2 + 2 v w u 2 - v 2 u + 4 v u + v 2 w u -4 v w u + w u - u -2 v + 2 v w - w + 1$ (db)
Jones polynomial	$- q 8 + 3 q 7 -5 q 6 + 7 q 5 -8 q 4 + 9 q 3 -6 q 2 + 6 q -2 + q -1$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -4 -2 z 4 a -2 + 4 z 4 a -4 - z 4 a -6 -5 z 2 a -2 + 6 z 2 a -4 -2 z 2 a -6 + z 2 -4 a -2 + 3 a -4 - a -6 + 2-2 a -2 z -2 + a -4 z -2 + z -2$ (db)
Kauffman polynomial	$z 9 a -3 + z 9 a -5 + z 8 a -2 + 4 z 8 a -4 + 3 z 8 a -6 -3 z 7 a -3 + z 7 a -5 + 4 z 7 a -7 -4 z 6 a -2 -14 z 6 a -4 -7 z 6 a -6 + 3 z 6 a -8 + 2 z 5 a -1 + 7 z 5 a -3 -6 z 5 a -5 -10 z 5 a -7 + z 5 a -9 + 14 z 4 a -2 + 27 z 4 a -4 + 7 z 4 a -6 -7 z 4 a -8 + z 4 -2 z 3 a -1 - z 3 a -3 + 9 z 3 a -5 + 6 z 3 a -7 -2 z 3 a -9 -17 z 2 a -2 -21 z 2 a -4 -5 z 2 a -6 + 2 z 2 a -8 -3 z 2 -2 z a -1 -4 z a -3 -3 z a -5 - z a -7 + 7 a -2 + 7 a -4 + 2 a -6 + 3 + 2 a -1 z -1 + 2 a -3 z -1 -2 a -2 z -2 - a -4 z -2 - z -2$ (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 =

2 is the signature of L11n364. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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