L11n362

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_18,6,19,5 X₈₄₉₃ X_9,21,10,20 X_19,11,20,10 X_14,18,15,17 X_22,16,17,15 X_16,22,5,21 X_2,12,3,11
Gauss code	{1, -11, 5, -3}, {8, -4, -7, 6, 10, -9}, {4, -1, 2, -5, -6, 7, 11, -2, 3, -8, 9, -10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 4 - v w u 4 + v 2 u 3 -3 v u 3 - v 2 w u 3 + 3 v w u 3 - w u 3 + u 3 -2 v 2 u 2 + 4 v u 2 + 2 v 2 w u 2 -4 v w u 2 + 2 w u 2 -2 u 2 + v 2 u -3 v u - v 2 w u + 3 v w u - w u + u + v - v w$ (db)
Jones polynomial	$3 q 7 -5 q 6 + 10 q 5 -12 q 4 + 14 q 3 -13 q 2 + 11 q -7 + 4 q -1 - q -2$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + z 6 a -4 + 2 z 4 a -2 + 3 z 4 a -4 - z 4 a -6 - z 4 + 4 z 2 a -4 -3 z 2 a -6 - z 2 - a -2 + 3 a -4 -4 a -6 + a -8 + 1 + a -4 z -2 -2 a -6 z -2 + a -8 z -2$ (db)
Kauffman polynomial	$2 z 9 a -3 + 2 z 9 a -5 + 5 z 8 a -2 + 10 z 8 a -4 + 5 z 8 a -6 + 6 z 7 a -1 + 6 z 7 a -3 + 3 z 7 a -5 + 3 z 7 a -7 -5 z 6 a -2 -25 z 6 a -4 -16 z 6 a -6 + 4 z 6 + a z 5 -10 z 5 a -1 -18 z 5 a -3 -13 z 5 a -5 -6 z 5 a -7 - z 4 a -2 + 33 z 4 a -4 + 33 z 4 a -6 + 6 z 4 a -8 -7 z 4 - a z 3 + 3 z 3 a -1 + 14 z 3 a -3 + 20 z 3 a -5 + 10 z 3 a -7 -3 z 2 a -2 -24 z 2 a -4 -33 z 2 a -6 -14 z 2 a -8 + 2 z 2 - z a -1 -3 z a -3 -12 z a -5 -10 z a -7 + 2 a -2 + 9 a -4 + 15 a -6 + 8 a -8 + 1 + 2 a -5 z -1 + 2 a -7 z -1 - a -4 z -2 -2 a -6 z -2 - a -8 z -2$ (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 L11n362. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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