L11n319

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_14,5,15,6 X_11,21,12,20 X_17,11,18,22 X_21,17,22,16 X_10,13,5,14 X_8,20,9,19 X_18,8,19,7 X_2,9,3,10 X_4,16,1,15
Gauss code	{1, -10, 2, -11}, {3, -1, 9, -8, 10, -7}, {-4, -2, 7, -3, 11, 6, -5, -9, 8, 4, -6, 5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 4 - v u 4 - v 2 w u 4 + v w u 4 - v 2 u 3 + 2 v u 3 + v 2 w u 3 -3 v w u 3 + w u 3 + v 2 u 2 -2 v u 2 + 2 v w u 2 - w u 2 - v 2 u + 3 v u -2 v w u + w u - u - v + v w - w + 1$ (db)
Jones polynomial	$-2 q 6 + 4 q 5 -7 q 4 + 10 q 3 -9 q 2 + 11 q -7 + 6 q -1 -3 q -2 + q -3$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$- z 8 a -2 -6 z 6 a -2 + z 6 a -4 + z 6 -13 z 4 a -2 + 5 z 4 a -4 + 4 z 4 -14 z 2 a -2 + 9 z 2 a -4 - z 2 a -6 + 5 z 2 -10 a -2 + 8 a -4 -2 a -6 + 4-5 a -2 z -2 + 4 a -4 z -2 - a -6 z -2 + 2 z -2$ (db)
Kauffman polynomial	$2 z 9 a -1 + 2 z 9 a -3 + 9 z 8 a -2 + 5 z 8 a -4 + 4 z 8 + 3 a z 7 - z 7 a -1 + 4 z 7 a -5 + a 2 z 6 -34 z 6 a -2 -19 z 6 a -4 + z 6 a -6 -13 z 6 -9 a z 5 -11 z 5 a -1 -15 z 5 a -3 -13 z 5 a -5 -3 a 2 z 4 + 50 z 4 a -2 + 35 z 4 a -4 + 2 z 4 a -6 + 14 z 4 + 5 a z 3 + 14 z 3 a -1 + 31 z 3 a -3 + 25 z 3 a -5 + 3 z 3 a -7 + 2 a 2 z 2 -37 z 2 a -2 -26 z 2 a -4 -4 z 2 a -6 -13 z 2 -11 z a -1 -24 z a -3 -18 z a -5 -5 z a -7 + 15 a -2 + 12 a -4 + 3 a -6 + 7 + 5 a -1 z -1 + 9 a -3 z -1 + 5 a -5 z -1 + a -7 z -1 -5 a -2 z -2 -4 a -4 z -2 - a -6 z -2 -2 z -2$ (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 L11n319. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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