L11n386

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_15,1,16,4 X_5,12,6,13 X₃₈₄₉ X_9,16,10,17 X_17,19,18,22 X_11,20,12,21 X_19,10,20,11 X_21,5,22,18 X_2,14,3,13
Gauss code	{1, -11, -5, 3}, {-9, 8, -10, 7}, {-4, -1, 2, 5, -6, 9, -8, 4, 11, -2, -3, 6, -7, 10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 3 - v w u 3 + w u 3 - u 3 -2 v u 2 + 2 v w u 2 -2 w u 2 + 2 u 2 + 2 v u -2 v w u + 2 w u -2 u - v + v w - w + 1$ (db)
Jones polynomial	$2 q -3 + 7 q -1 -6 q -2 + 9 q -3 -8 q -4 + 6 q -5 -4 q -6 + 2 q -7 - q -8$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- z 4 a 6 -3 z 2 a 6 - a 6 z -2 -3 a 6 + z 6 a 4 + 5 z 4 a 4 + 11 z 2 a 4 + 4 a 4 z -2 + 10 a 4 -3 z 4 a 2 -10 z 2 a 2 -5 a 2 z -2 -11 a 2 + 2 z 2 + 2 z -2 + 4$ (db)
Kauffman polynomial	$z 5 a 9 -3 z 3 a 9 + z a 9 + 2 z 6 a 8 -5 z 4 a 8 + z 2 a 8 + 3 z 7 a 7 -9 z 5 a 7 + 9 z 3 a 7 -5 z a 7 + a 7 z -1 + 3 z 8 a 6 -10 z 6 a 6 + 14 z 4 a 6 -9 z 2 a 6 - a 6 z -2 + 5 a 6 + z 9 a 5 + 2 z 7 a 5 -17 z 5 a 5 + 35 z 3 a 5 -22 z a 5 + 5 a 5 z -1 + 5 z 8 a 4 -20 z 6 a 4 + 39 z 4 a 4 -34 z 2 a 4 -4 a 4 z -2 + 16 a 4 + z 9 a 3 -8 z 5 a 3 + 26 z 3 a 3 -26 z a 3 + 9 a 3 z -1 + 2 z 8 a 2 -8 z 6 a 2 + 23 z 4 a 2 -32 z 2 a 2 -5 a 2 z -2 + 17 a 2 + z 7 a - z 5 a + 3 z 3 a -10 z a + 5 a z -1 + 3 z 4 -8 z 2 -2 z -2 + 7$ (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 L11n386. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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