L11a343

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{(t(1)-1) (t(2)-1) \left(t(2)^2 t(1)^2-2 t(2) t(1)^2+t(1)^2-2 t(2)^2 t(1)+3 t(2) t(1)-2 t(1)+t(2)^2-2 t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db)
Jones polynomial	$q^{3/2}-4 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{14}{q^{3/2}}+\frac{16}{q^{5/2}}-\frac{20}{q^{7/2}}+\frac{19}{q^{9/2}}-\frac{16}{q^{11/2}}+\frac{12}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a 9 z + a 9 z -1 -3 a 7 z 3 -7 a 7 z -5 a 7 z -1 + 3 a 5 z 5 + 10 a 5 z 3 + 14 a 5 z + 8 a 5 z -1 - a 3 z 7 -4 a 3 z 5 -8 a 3 z 3 -9 a 3 z -4 a 3 z -1 + a z 5 + 2 a z 3 + a z$ (db)
Kauffman polynomial	$a 11 z 5 -2 a 11 z 3 + a 11 z + 3 a 10 z 6 -6 a 10 z 4 + 5 a 10 z 2 -2 a 10 + 4 a 9 z 7 -3 a 9 z 5 -2 a 9 z 3 + a 9 z + a 9 z -1 + 4 a 8 z 8 + 3 a 8 z 6 -18 a 8 z 4 + 22 a 8 z 2 -9 a 8 + 3 a 7 z 9 + 6 a 7 z 7 -17 a 7 z 5 + 16 a 7 z 3 -9 a 7 z + 5 a 7 z -1 + a 6 z 10 + 10 a 6 z 8 -13 a 6 z 6 -12 a 6 z 4 + 29 a 6 z 2 -14 a 6 + 7 a 5 z 9 -27 a 5 z 5 + 33 a 5 z 3 -21 a 5 z + 8 a 5 z -1 + a 4 z 10 + 12 a 4 z 8 -26 a 4 z 6 + 5 a 4 z 4 + 13 a 4 z 2 -8 a 4 + 4 a 3 z 9 + 2 a 3 z 7 -24 a 3 z 5 + 25 a 3 z 3 -14 a 3 z + 4 a 3 z -1 + 6 a 2 z 8 -12 a 2 z 6 + 3 a 2 z 4 + 2 a 2 z 2 + 4 a z 7 -10 a z 5 + 8 a z 3 -2 a z + z 6 -2 z 4 + z 2$ (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 =

-3 is the signature of L11a343. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$
$r = -8$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -5$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -4$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -3$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = -1$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 0$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{9}$
$r = 1$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$