L11n363

(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_5,18,6,19 X₈₄₉₃ X_9,21,10,20 X_19,11,20,10 X_15,22,16,17 X_17,16,18,5 X_21,14,22,15 X_2,12,3,11
Gauss code	{1, -11, 5, -3}, {-9, 4, -7, 6, -10, 8}, {-4, -1, 2, -5, -6, 7, 11, -2, 3, 10, -8, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 2 + u 2 + 2 v u -2 u - v + 1$ (db)
Jones polynomial	$q 2 - q + 1 + q -1 - q -2 + 4 q -3 -3 q -4 + 4 q -5 -3 q -6 + 2 q -7 - q -8$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- z 4 a 6 -3 z 2 a 6 -3 a 6 + z 6 a 4 + 6 z 4 a 4 + 13 z 2 a 4 + a 4 z -2 + 9 a 4 - z 6 a 2 -7 z 4 a 2 -14 z 2 a 2 -2 a 2 z -2 -10 a 2 + z 4 + 4 z 2 + z -2 + 4$ (db)
Kauffman polynomial	$z 5 a 9 -3 z 3 a 9 + z a 9 + 2 z 6 a 8 -6 z 4 a 8 + 3 z 2 a 8 - a 8 + 2 z 7 a 7 -6 z 5 a 7 + 3 z 3 a 7 + z 8 a 6 -3 z 6 a 6 + 3 z 4 a 6 -3 z 2 a 6 + 2 a 6 + 2 z 7 a 5 -9 z 5 a 5 + 14 z 3 a 5 -6 z a 5 + 2 z 8 a 4 -15 z 6 a 4 + 37 z 4 a 4 -34 z 2 a 4 - a 4 z -2 + 13 a 4 + z 9 a 3 -7 z 7 a 3 + 11 z 5 a 3 + 3 z 3 a 3 -9 z a 3 + 2 a 3 z -1 + 2 z 8 a 2 -17 z 6 a 2 + 43 z 4 a 2 -41 z 2 a 2 -2 a 2 z -2 + 15 a 2 + z 9 a -7 z 7 a + 13 z 5 a -5 z 3 a -4 z a + 2 a z -1 + z 8 -7 z 6 + 15 z 4 -13 z 2 - z -2 + 6$ (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 L11n363. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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