L11n358

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,19,12,18 X_7,16,8,17 X_15,8,16,9 X_17,15,18,22 X_21,13,22,12 X_13,21,14,20 X_19,5,20,14 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {-5, 4, -6, 3, -9, 8, -7, 6}, {10, -1, -4, 5, 11, -2, -3, 7, -8, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$u v 4 - v 4 + u 2 v 3 -3 u v 3 + v 3 - u 2 v 2 + 2 u v 2 -2 u w v 2 + w v 2 - u 2 w v + 3 u w v - w v + u 2 w - u w$ (db)
Jones polynomial	$- q 3 + 3 q 2 -4 q + 7-6 q -1 + 7 q -2 -5 q -3 + 4 q -4 -2 q -5 + q -6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a 6 -2 z 2 a 4 + a 4 z -2 - a 4 + z 4 a 2 -2 a 2 z -2 -2 a 2 + z 4 + z 2 + z -2 + 2- z 2 a -2$ (db)
Kauffman polynomial	$a 3 z 9 + a z 9 + 2 a 4 z 8 + 5 a 2 z 8 + 3 z 8 + 2 a 5 z 7 + 2 z 7 a -1 + a 6 z 6 -6 a 4 z 6 -22 a 2 z 6 -15 z 6 -7 a 5 z 5 -10 a 3 z 5 -11 a z 5 -8 z 5 a -1 -4 a 6 z 4 + 4 a 4 z 4 + 37 a 2 z 4 + 3 z 4 a -2 + 32 z 4 + 5 a 5 z 3 + 15 a 3 z 3 + 24 a z 3 + 15 z 3 a -1 + z 3 a -3 + 4 a 6 z 2 -5 a 4 z 2 -30 a 2 z 2 -5 z 2 a -2 -26 z 2 - a 5 z -10 a 3 z -14 a z -7 z a -1 -2 z a -3 - a 6 + 4 a 4 + 12 a 2 + 2 a -2 + 10 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 =

0 is the signature of L11n358. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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