L11n336

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_16,12,17,11 X₈₄₉₃ X_2,18,3,17 X_14,6,15,5 X_18,7,19,8 X_12,16,5,15 X_13,20,14,21 X_9,13,10,22 X_21,11,22,10 X_4,19,1,20
Gauss code	{1, -4, 3, -11}, {5, -1, 6, -3, -9, 10, 2, -7}, {-8, -5, 7, -2, 4, -6, 11, 8, -10, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 3 - v w u 3 + w u 3 - u 3 + v 2 u 2 -2 v u 2 - v 2 w u 2 + 2 v w u 2 - w u 2 + u 2 + v 3 u -2 v 2 u + v u - v 3 w u + 2 v 2 w u - v w u - v 3 + v 2 + v 3 w - v 2 w$ (db)
Jones polynomial	$- q 7 + 3 q 6 -4 q 5 + 8 q 4 -7 q 3 + 8 q 2 -7 q + 6-3 q -1 + q -2$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$2 z 4 a -2 + z 4 a -4 + 4 z 2 a -2 - z 2 a -6 -3 z 2 + a 2 + 3 a -2 -2 a -4 -2 + a -2 z -2 -2 a -4 z -2 + a -6 z -2$ (db)
Kauffman polynomial	$2 z 9 a -3 + 2 z 9 a -5 + 5 z 8 a -2 + 8 z 8 a -4 + 3 z 8 a -6 + 5 z 7 a -1 - z 7 a -3 -5 z 7 a -5 + z 7 a -7 -17 z 6 a -2 -33 z 6 a -4 -14 z 6 a -6 + 2 z 6 -17 z 5 a -1 -16 z 5 a -3 -3 z 5 a -5 -4 z 5 a -7 + 15 z 4 a -2 + 39 z 4 a -4 + 20 z 4 a -6 -4 z 4 + 3 a z 3 + 19 z 3 a -1 + 18 z 3 a -3 + 6 z 3 a -5 + 4 z 3 a -7 + a 2 z 2 -3 z 2 a -2 -17 z 2 a -4 -9 z 2 a -6 + 6 z 2 -2 a z -6 z a -1 -2 z a -3 + 2 z a -5 - a 2 - a -2 -2 a -4 -2 a -6 -1-2 a -3 z -1 -2 a -5 z -1 + a -2 z -2 + 2 a -4 z -2 + a -6 z -2$ (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 =

0 is the signature of L11n336. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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