L11n340

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_18,12,19,11 X_7,14,8,15 X_13,8,14,9 X_22,20,13,19 X_20,16,21,15 X_16,22,17,21 X_12,18,5,17 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, 3, -9}, {-5, 4, 7, -8, 9, -3, 6, -7, 8, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 3 - v u 3 - v 2 w u 3 + v w u 3 + v 3 u 2 -4 v 2 u 2 + 3 v u 2 + 3 v 2 w u 2 -4 v w u 2 + w u 2 - v 3 u + 4 v 2 u -3 v u -3 v 2 w u + 4 v w u - w u - v 2 + v + v 2 w - v w$ (db)
Jones polynomial	$- q 5 + 3 q 4 -7 q 3 + 11 q 2 -13 q + 14-12 q -1 + 11 q -2 -5 q -3 + 3 q -4$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 6 + a 2 z 4 + 2 z 4 a -2 -2 z 4 -2 a 2 z 2 + 3 z 2 a -2 - z 2 a -4 + 2 a 4 -7 a 2 + a -2 - a -4 + 5 + 2 a 4 z -2 -5 a 2 z -2 - a -2 z -2 + 4 z -2$ (db)
Kauffman polynomial	$a z 9 + z 9 a -1 + 4 a 2 z 8 + 4 z 8 a -2 + 8 z 8 + 3 a 3 z 7 + 11 a z 7 + 13 z 7 a -1 + 5 z 7 a -3 -8 a 2 z 6 + 3 z 6 a -4 -11 z 6 -6 a 3 z 5 -32 a z 5 -35 z 5 a -1 -8 z 5 a -3 + z 5 a -5 + 6 a 4 z 4 + 18 a 2 z 4 -10 z 4 a -2 -5 z 4 a -4 + 7 z 4 + 14 a 3 z 3 + 43 a z 3 + 36 z 3 a -1 + 5 z 3 a -3 -2 z 3 a -5 -13 a 4 z 2 -26 a 2 z 2 + 6 z 2 a -2 + 3 z 2 a -4 -10 z 2 -16 a 3 z -32 a z -20 z a -1 -3 z a -3 + z a -5 + 9 a 4 + 18 a 2 - a -4 + 11 + 5 a 3 z -1 + 9 a z -1 + 5 a -1 z -1 + a -3 z -1 -2 a 4 z -2 -5 a 2 z -2 - a -2 z -2 -4 z -2$ (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 L11n340. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$
$r = -4$	${\mathbb Z}^{3}$	${\mathbb Z}^{2}$
$r = -3$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -1$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 0$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{7}$
$r = 1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 2$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 3$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$