L11a442

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,3,15,4 X_22,16,13,15 X_8,18,9,17 X_16,8,17,7 X_18,10,19,9 X_20,12,21,11 X_10,20,11,19 X_12,22,5,21 X₂₅₃₆ X_4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 5, -4, 6, -8, 7, -9}, {11, -2, 3, -5, 4, -6, 8, -7, 9, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v 3 u 3 - v 2 u 3 - v 3 w u 3 + v 2 w u 3 - v 3 u 2 + 2 v 2 u 2 - v u 2 + v 3 w u 2 -2 v 2 w u 2 + v w u 2 - v 2 u + 2 v u + v 2 w u -2 v w u + w u - u - v + v w -2 w + 1$ (db)
Jones polynomial	$- q 9 + 3 q 8 -4 q 7 + 6 q 6 -7 q 5 + 8 q 4 -7 q 3 + 6 q 2 -4 q + 4- q -1 + q -2$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z 8 a -4 -2 z 6 a -2 + 6 z 6 a -4 - z 6 a -6 -11 z 4 a -2 + 12 z 4 a -4 -4 z 4 a -6 + z 4 -18 z 2 a -2 + 11 z 2 a -4 -3 z 2 a -6 + 5 z 2 -11 a -2 + 5 a -4 + 6-2 a -2 z -2 + a -4 z -2 + z -2$ (db)
Kauffman polynomial	$z 10 a -2 + z 10 a -4 + z 9 a -1 + 4 z 9 a -3 + 3 z 9 a -5 -3 z 8 a -2 + 4 z 8 a -6 + z 8 -4 z 7 a -1 -18 z 7 a -3 -10 z 7 a -5 + 4 z 7 a -7 -5 z 6 a -2 -13 z 6 a -4 -11 z 6 a -6 + 4 z 6 a -8 -7 z 6 + 18 z 5 a -3 + 8 z 5 a -5 -6 z 5 a -7 + 4 z 5 a -9 + 22 z 4 a -2 + 19 z 4 a -4 + 8 z 4 a -6 -3 z 4 a -8 + 3 z 4 a -10 + 17 z 4 + 12 z 3 a -1 + 6 z 3 a -3 -2 z 3 a -5 -3 z 3 a -9 + z 3 a -11 -25 z 2 a -2 -10 z 2 a -4 -3 z 2 a -6 -2 z 2 a -8 -2 z 2 a -10 -18 z 2 -11 z a -1 -11 z a -3 + 13 a -2 + 5 a -4 + a -8 + 8 + 2 a -1 z -1 + 2 a -3 z -1 -2 a -2 z -2 - a -4 z -2 - z -2$ (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 =

4 is the signature of L11a442. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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