L11n329

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_5,14,6,15 X₃₈₄₉ X_15,2,16,3 X_16,7,17,8 X_13,18,14,19 X_9,13,10,22 X_11,21,12,20 X_19,5,20,12 X_21,11,22,10 X_4,17,1,18
Gauss code	{1, 4, -3, -11}, {-2, -1, 5, 3, -7, 10, -8, 9}, {-6, 2, -4, -5, 11, 6, -9, 8, -10, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 w u 3 + v w u 3 - v 3 w u 2 + v 2 w u 2 - u 2 - v u + v 3 w u + u - v 2 + v$ (db)
Jones polynomial	$q 2 - q + 2-2 q -1 + 4 q -2 -2 q -3 + 3 q -4 -2 q -5 + 2 q -6 - q -7$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- a 8 + z 4 a 6 + 4 z 2 a 6 + 2 a 6 - z 6 a 4 -4 z 4 a 4 -2 z 2 a 4 + a 4 z -2 + a 4 - z 6 a 2 -5 z 4 a 2 -7 z 2 a 2 -2 a 2 z -2 -5 a 2 + z 4 + 4 z 2 + z -2 + 3$ (db)
Kauffman polynomial	$z a 9 + 2 z 2 a 8 - a 8 + z 5 a 7 -2 z 3 a 7 + z a 7 + 2 z 6 a 6 -7 z 4 a 6 + 5 z 2 a 6 + 2 z 7 a 5 -8 z 5 a 5 + 8 z 3 a 5 -3 z a 5 + 2 z 8 a 4 -10 z 6 a 4 + 17 z 4 a 4 -18 z 2 a 4 - a 4 z -2 + 9 a 4 + z 9 a 3 -4 z 7 a 3 + z 5 a 3 + 8 z 3 a 3 -8 z a 3 + 2 a 3 z -1 + 3 z 8 a 2 -19 z 6 a 2 + 40 z 4 a 2 -35 z 2 a 2 -2 a 2 z -2 + 13 a 2 + z 9 a -6 z 7 a + 10 z 5 a -2 z 3 a -5 z a + 2 a z -1 + z 8 -7 z 6 + 16 z 4 -14 z 2 - z -2 + 6$ (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 =

-4 is the signature of L11n329. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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