L11a334

(Knotscape image)

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

Planar diagram presentation	X_10,1,11,2 X_20,6,21,5 X_14,3,15,4 X_4,15,5,16 X_22,20,9,19 X_16,7,17,8 X_18,12,19,11 X_12,18,13,17 X_2,9,3,10 X_8,13,1,14 X_6,22,7,21
Gauss code	{1, -9, 3, -4, 2, -11, 6, -10}, {9, -1, 7, -8, 10, -3, 4, -6, 8, -7, 5, -2, 11, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{t(2)^3 t(1)^3-3 t(2)^2 t(1)^3+2 t(2) t(1)^3-4 t(2)^3 t(1)^2+11 t(2)^2 t(1)^2-11 t(2) t(1)^2+3 t(1)^2+3 t(2)^3 t(1)-11 t(2)^2 t(1)+11 t(2) t(1)-4 t(1)+2 t(2)^2-3 t(2)+1}{t(1)^{3/2} t(2)^{3/2}}$ (db)
Jones polynomial	$q^{9/2}-\frac{8}{q^{9/2}}-4 q^{7/2}+\frac{14}{q^{7/2}}+9 q^{5/2}-\frac{20}{q^{5/2}}-15 q^{3/2}+\frac{22}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{3}{q^{11/2}}+20 \sqrt{q}-\frac{23}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 -2 a 3 z 5 + 3 a z 5 -2 z 5 a -1 + a 5 z 3 -5 a 3 z 3 + 4 a z 3 -4 z 3 a -1 + z 3 a -3 + 2 a 5 z -5 a 3 z + 2 a z -2 z a -1 + z a -3 + a 5 z -1 - a 3 z -1$ (db)
Kauffman polynomial	$-2 a 2 z 10 -2 z 10 -6 a 3 z 9 -12 a z 9 -6 z 9 a -1 -8 a 4 z 8 -14 a 2 z 8 -7 z 8 a -2 -13 z 8 -6 a 5 z 7 + 15 a z 7 + 5 z 7 a -1 -4 z 7 a -3 -3 a 6 z 6 + 12 a 4 z 6 + 35 a 2 z 6 + 15 z 6 a -2 - z 6 a -4 + 36 z 6 - a 7 z 5 + 9 a 5 z 5 + 16 a 3 z 5 + 7 a z 5 + 10 z 5 a -1 + 9 z 5 a -3 + 4 a 6 z 4 -10 a 4 z 4 -27 a 2 z 4 -9 z 4 a -2 + 2 z 4 a -4 -24 z 4 + 2 a 7 z 3 -8 a 5 z 3 -21 a 3 z 3 -13 a z 3 -8 z 3 a -1 -6 z 3 a -3 - a 6 z 2 + 2 a 4 z 2 + 6 a 2 z 2 + 2 z 2 a -2 - z 2 a -4 + 6 z 2 - a 7 z + 5 a 5 z + 9 a 3 z + 4 a z + 2 z a -1 + z a -3 + a 4 - a 5 z -1 - a 3 z -1$ (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 L11a334. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = -1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{12}$
$r = 1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$