L11a344

(Knotscape image)

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

Planar diagram presentation	X_12,1,13,2 X_18,7,19,8 X_10,5,1,6 X₆₃₇₄ X_4,9,5,10 X_20,14,21,13 X_22,16,11,15 X_14,22,15,21 X_16,20,17,19 X_2,11,3,12 X_8,17,9,18
Gauss code	{1, -10, 4, -5, 3, -4, 2, -11, 5, -3}, {10, -1, 6, -8, 7, -9, 11, -2, 9, -6, 8, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + 4 v 2 u 3 -4 v u 3 + 2 u 3 + 2 v 3 u 2 -6 v 2 u 2 + 6 v u 2 -2 u 2 -2 v 3 u + 6 v 2 u -6 v u + 2 u + 2 v 3 -4 v 2 + 4 v -1$ (db)
Jones polynomial	$-q^{7/2}+4 q^{5/2}-8 q^{3/2}+13 \sqrt{q}-\frac{16}{\sqrt{q}}+\frac{17}{q^{3/2}}-\frac{18}{q^{5/2}}+\frac{13}{q^{7/2}}-\frac{10}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{2}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 -2 a 7 z -1 + 3 z 3 a 5 + 9 z a 5 + 7 a 5 z -1 -3 z 5 a 3 -11 z 3 a 3 -15 z a 3 -7 a 3 z -1 + z 7 a + 4 z 5 a + 7 z 3 a + 6 z a + 2 a z -1 - z 5 a -1 -2 z 3 a -1 - z a -1$ (db)
Kauffman polynomial	$- a 4 z 10 - a 2 z 10 -2 a 5 z 9 -6 a 3 z 9 -4 a z 9 -2 a 6 z 8 -4 a 4 z 8 -9 a 2 z 8 -7 z 8 -2 a 7 z 7 - a 5 z 7 + 6 a 3 z 7 -2 a z 7 -7 z 7 a -1 - a 8 z 6 + 5 a 4 z 6 + 16 a 2 z 6 -4 z 6 a -2 + 8 z 6 + 6 a 7 z 5 + 11 a 5 z 5 + 5 a 3 z 5 + 12 a z 5 + 11 z 5 a -1 - z 5 a -3 + 4 a 8 z 4 + 13 a 6 z 4 + 11 a 4 z 4 -2 a 2 z 4 + 6 z 4 a -2 + 2 z 4 -6 a 7 z 3 -17 a 5 z 3 -15 a 3 z 3 -9 a z 3 -4 z 3 a -1 + z 3 a -3 -5 a 8 z 2 -18 a 6 z 2 -24 a 4 z 2 -13 a 2 z 2 -2 z 2 a -2 -4 z 2 + 4 a 7 z + 16 a 5 z + 15 a 3 z + 4 a z + z a -1 + 2 a 8 + 8 a 6 + 13 a 4 + 8 a 2 + 2-2 a 7 z -1 -7 a 5 z -1 -7 a 3 z -1 -2 a z -1$ (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 =

-1 is the signature of L11a344. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{5}$
$r = -3$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -1$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 0$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{8}$
$r = 1$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$