L11a342

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 3 + 4 v u 3 - u 3 -2 v 3 u 2 + 10 v 2 u 2 -10 v u 2 + 2 u 2 + 2 v 3 u -10 v 2 u + 10 v u -2 u - v 3 + 4 v 2 -2 v$ (db)
Jones polynomial	$-\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{13}{q^{5/2}}-\frac{18}{q^{7/2}}+\frac{20}{q^{9/2}}-\frac{20}{q^{11/2}}+\frac{17}{q^{13/2}}-\frac{13}{q^{15/2}}+\frac{7}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a 11 z -1 + 4 z a 9 + 4 a 9 z -1 -6 z 3 a 7 -11 z a 7 -6 a 7 z -1 + 3 z 5 a 5 + 8 z 3 a 5 + 10 z a 5 + 5 a 5 z -1 + z 5 a 3 - z 3 a 3 -4 z a 3 -2 a 3 z -1 - z 3 a - z a$ (db)
Kauffman polynomial	$- z 6 a 12 + 3 z 4 a 12 -3 z 2 a 12 + a 12 -3 z 7 a 11 + 8 z 5 a 11 -8 z 3 a 11 + 4 z a 11 - a 11 z -1 -4 z 8 a 10 + 5 z 6 a 10 + 5 z 4 a 10 -9 z 2 a 10 + 3 a 10 -3 z 9 a 9 -5 z 7 a 9 + 27 z 5 a 9 -31 z 3 a 9 + 18 z a 9 -4 a 9 z -1 - z 10 a 8 -11 z 8 a 8 + 22 z 6 a 8 -4 z 4 a 8 -7 z 2 a 8 + 3 a 8 -7 z 9 a 7 -4 z 7 a 7 + 41 z 5 a 7 -52 z 3 a 7 + 29 z a 7 -6 a 7 z -1 - z 10 a 6 -14 z 8 a 6 + 28 z 6 a 6 -15 z 4 a 6 + a 6 -4 z 9 a 5 -8 z 7 a 5 + 33 z 5 a 5 -41 z 3 a 5 + 23 z a 5 -5 a 5 z -1 -7 z 8 a 4 + 9 z 6 a 4 -5 z 4 a 4 + a 4 -6 z 7 a 3 + 10 z 5 a 3 -10 z 3 a 3 + 7 z a 3 -2 a 3 z -1 -3 z 6 a 2 + 4 z 4 a 2 - z 2 a 2 - z 5 a + 2 z 3 a - z a$ (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 L11a342. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$
$r = -9$	${\mathbb Z}$
$r = -8$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -6$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -5$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -4$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{10}$
$r = -3$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -1$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 0$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{6}$
$r = 1$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 2$	${\mathbb Z}_2$	${\mathbb Z}$