L11n434

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_14,4,15,3 X_12,14,7,13 X₂₇₃₈ X_22,10,13,9 X_6,22,1,21 X_15,20,16,21 X_5,17,6,16 X_18,11,19,12 X_10,17,11,18 X_19,5,20,4
Gauss code	{1, -4, 2, 11, -8, -6}, {4, -1, 5, -10, 9, -3}, {3, -2, -7, 8, 10, -9, -11, 7, 6, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 3 - v u 3 - v 2 w u 3 + 2 v w u 3 - w u 3 - v 2 u 2 + 2 v u 2 + v 2 w u 2 -3 v w u 2 + w u 2 -2 v w 2 u + w 2 u - v 2 w u + 3 v w u - w u + v w 2 - w 2 + v 2 w -2 v w + w$ (db)
Jones polynomial	$q 6 -3 q 5 + 6 q 4 -7 q 3 + 10 q 2 -9 q + 9-6 q -1 + 4 q -2 - q -3$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 6 a -2 -4 z 4 a -2 + z 4 a -4 + 2 z 4 - a 2 z 2 -7 z 2 a -2 + 2 z 2 a -4 + 4 z 2 -5 a -2 + 2 a -4 + 3-2 a -2 z -2 + a -4 z -2 + z -2$ (db)
Kauffman polynomial	$2 z 9 a -1 + 2 z 9 a -3 + 8 z 8 a -2 + 4 z 8 a -4 + 4 z 8 + 2 a z 7 -3 z 7 a -1 -2 z 7 a -3 + 3 z 7 a -5 -30 z 6 a -2 -13 z 6 a -4 + z 6 a -6 -16 z 6 -4 a z 5 -2 z 5 a -1 -7 z 5 a -3 -9 z 5 a -5 + 4 a 2 z 4 + 44 z 4 a -2 + 13 z 4 a -4 -3 z 4 a -6 + 32 z 4 + a 3 z 3 + 8 a z 3 + 10 z 3 a -1 + 8 z 3 a -3 + 5 z 3 a -5 -4 a 2 z 2 -33 z 2 a -2 -11 z 2 a -4 + 2 z 2 a -6 -24 z 2 - a 3 z -3 a z -8 z a -1 -6 z a -3 + a 2 + 12 a -2 + 6 a -4 + 8 + 2 a -1 z -1 + 2 a -3 z -1 -2 a -2 z -2 - a -4 z -2 - z -2$ (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 L11n434. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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