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 $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + 3 v 2 u 3 -3 v u 3 + u 3 + 3 v 3 u 2 -8 v 2 u 2 + 8 v u 2 -3 u 2 -3 v 3 u + 8 v 2 u -8 v u + 3 u + v 3 -3 v 2 + 3 v -1$ (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	$z a 9 + a 9 z -1 -3 z 3 a 7 -7 z a 7 -5 a 7 z -1 + 3 z 5 a 5 + 10 z 3 a 5 + 14 z a 5 + 8 a 5 z -1 - z 7 a 3 -4 z 5 a 3 -8 z 3 a 3 -9 z a 3 -4 a 3 z -1 + z 5 a + 2 z 3 a + z a$ (db)
Kauffman polynomial	$- z 5 a 11 + 2 z 3 a 11 - z a 11 -3 z 6 a 10 + 6 z 4 a 10 -5 z 2 a 10 + 2 a 10 -4 z 7 a 9 + 3 z 5 a 9 + 2 z 3 a 9 - z a 9 - a 9 z -1 -4 z 8 a 8 -3 z 6 a 8 + 18 z 4 a 8 -22 z 2 a 8 + 9 a 8 -3 z 9 a 7 -6 z 7 a 7 + 17 z 5 a 7 -16 z 3 a 7 + 9 z a 7 -5 a 7 z -1 - z 10 a 6 -10 z 8 a 6 + 13 z 6 a 6 + 12 z 4 a 6 -29 z 2 a 6 + 14 a 6 -7 z 9 a 5 + 27 z 5 a 5 -33 z 3 a 5 + 21 z a 5 -8 a 5 z -1 - z 10 a 4 -12 z 8 a 4 + 26 z 6 a 4 -5 z 4 a 4 -13 z 2 a 4 + 8 a 4 -4 z 9 a 3 -2 z 7 a 3 + 24 z 5 a 3 -25 z 3 a 3 + 14 z a 3 -4 a 3 z -1 -6 z 8 a 2 + 12 z 6 a 2 -3 z 4 a 2 -2 z 2 a 2 -4 z 7 a + 10 z 5 a -8 z 3 a + 2 z a - z 6 + 2 z 4 - z 2$ (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.

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