L11n405

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_16,9,17,10 X_8,15,9,16 X_4,17,1,18 X_22,12,19,11 X_10,4,11,3 X_5,21,6,20 X_21,5,22,18 X_12,20,13,19 X_2,14,3,13
Gauss code	{1, -11, 7, -5}, {10, 8, -9, -6}, {-8, -1, 2, -4, 3, -7, 6, -10, 11, -2, 4, -3, 5, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 5 - v w u 5 -2 v u 4 + 2 v w u 4 - w u 4 + u 4 + 2 v u 3 -2 v w u 3 + 2 w u 3 -2 u 3 -2 v u 2 + 2 v w u 2 -2 w u 2 + 2 u 2 + v u - v w u + 2 w u -2 u - w + 1$ (db)
Jones polynomial	$3 q 4 -4 q 3 + 8 q 2 -9 q + 12-10 q -1 + 8 q -2 -6 q -3 + 3 q -4 - q -5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z 8 - a 2 z 6 - z 6 a -2 + 6 z 6 -4 a 2 z 4 -5 z 4 a -2 + 13 z 4 -5 a 2 z 2 -10 z 2 a -2 + z 2 a -4 + 14 z 2 -3 a 2 -11 a -2 + 3 a -4 + 11- a 2 z -2 -5 a -2 z -2 + 2 a -4 z -2 + 4 z -2$ (db)
Kauffman polynomial	$2 a z 9 + 2 z 9 a -1 + 4 a 2 z 8 + 5 z 8 a -2 + 9 z 8 + 4 a 3 z 7 - z 7 a -1 + 3 z 7 a -3 + 3 a 4 z 6 -9 a 2 z 6 -22 z 6 a -2 -34 z 6 + a 5 z 5 -7 a 3 z 5 -11 a z 5 -12 z 5 a -1 -9 z 5 a -3 -6 a 4 z 4 + 10 a 2 z 4 + 46 z 4 a -2 + 6 z 4 a -4 + 56 z 4 -2 a 5 z 3 + a 3 z 3 + 23 a z 3 + 34 z 3 a -1 + 14 z 3 a -3 + a 4 z 2 -9 a 2 z 2 -45 z 2 a -2 -17 z 2 a -4 -38 z 2 + a 5 z -2 a 3 z -17 a z -29 z a -1 -15 z a -3 + 4 a 2 + 22 a -2 + 10 a -4 + 17 + a 3 z -1 + 5 a z -1 + 9 a -1 z -1 + 5 a -3 z -1 - a 2 z -2 -5 a -2 z -2 -2 a -4 z -2 -4 z -2$ (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 L11n405. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 0$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{8}$
$r = 1$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 4$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}^{3}$