L11n388

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_5,12,6,13 X₈₄₉₃ X_22,11,19,12 X_18,22,5,21 X_20,10,21,9 X_10,17,11,18 X_16,19,17,20 X_2,14,3,13
Gauss code	{1, -11, 5, -3}, {10, -8, 7, -6}, {-4, -1, 2, -5, 8, -9, 6, 4, 11, -2, 3, -10, 9, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 3 - v w u 3 + w u 3 - u 3 -3 v u 2 + 3 v w u 2 -3 w u 2 + 3 u 2 + 3 v u -3 v w u + 3 w u -3 u - v + v w - w + 1$ (db)
Jones polynomial	$- q 2 + 5 q -7 + 11 q -1 -10 q -2 + 12 q -3 -9 q -4 + 6 q -5 -3 q -6$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- a 6 z -2 -2 a 6 - z 4 a 4 + 2 z 2 a 4 + 4 a 4 z -2 + 7 a 4 + z 6 a 2 + 2 z 4 a 2 -2 z 2 a 2 -5 a 2 z -2 -8 a 2 - z 4 + 2 z -2 + 3$ (db)
Kauffman polynomial	$3 a 4 z 8 + 3 a 2 z 8 + 6 a 5 z 7 + 13 a 3 z 7 + 7 a z 7 + 3 a 6 z 6 + 3 a 4 z 6 + 5 a 2 z 6 + 5 z 6 -13 a 5 z 5 -27 a 3 z 5 -13 a z 5 + z 5 a -1 -8 a 4 z 4 -16 a 2 z 4 -8 z 4 + 6 a 7 z 3 + 24 a 5 z 3 + 24 a 3 z 3 + 6 a z 3 -3 a 4 z 2 -3 a 2 z 2 -6 a 7 z -19 a 5 z -21 a 3 z -8 a z + 3 a 6 + 10 a 4 + 11 a 2 + 5 + a 7 z -1 + 5 a 5 z -1 + 9 a 3 z -1 + 5 a z -1 - a 6 z -2 -4 a 4 z -2 -5 a 2 z -2 -2 z -2$ (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 L11n388. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -3$	$i = -1$
$r = -5$	${\mathbb Z}^{3}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -1$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 0$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{8}$
$r = 1$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$