L11a403

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_16,9,17,10 X_14,8,15,7 X_18,12,19,11 X_20,15,21,16 X_22,18,11,17 X_10,19,5,20 X_8,22,9,21 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, 3, -8}, {5, -2, 11, -4, 6, -3, 7, -5, 8, -6, 9, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{u v^2 w^4-2 u v^2 w^3+2 u v^2 w^2-u v^2 w-2 u v w^4+5 u v w^3-5 u v w^2+4 u v w-u v+u w^4-3 u w^3+4 u w^2-3 u w+u-v^2 w^4+3 v^2 w^3-4 v^2 w^2+3 v^2 w-v^2+v w^4-4 v w^3+5 v w^2-5 v w+2 v+w^3-2 w^2+2 w-1}{\sqrt{u} v w^2}$ (db)
Jones polynomial	$q 7 -4 q 6 + 9 q 5 -15 q 4 + 20 q 3 -22 q 2 + 23 q -18 + 15 q -1 -8 q -2 + 4 q -3 - q -4$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -4 + 3 z 4 a -4 + 3 z 2 a -4 + a -4 - z 8 a -2 -5 z 6 a -2 - a 2 z 4 -10 z 4 a -2 -2 a 2 z 2 -9 z 2 a -2 + a 2 z -2 + a -2 z -2 -2 a -2 + 2 z 6 + 7 z 4 + 7 z 2 -2 z -2 + 1$ (db)
Kauffman polynomial	$2 z 10 a -2 + 2 z 10 + 5 a z 9 + 14 z 9 a -1 + 9 z 9 a -3 + 4 a 2 z 8 + 23 z 8 a -2 + 15 z 8 a -4 + 12 z 8 + a 3 z 7 -11 a z 7 -28 z 7 a -1 -2 z 7 a -3 + 14 z 7 a -5 -14 a 2 z 6 -77 z 6 a -2 -25 z 6 a -4 + 9 z 6 a -6 -57 z 6 -3 a 3 z 5 -2 a z 5 -6 z 5 a -1 -30 z 5 a -3 -19 z 5 a -5 + 4 z 5 a -7 + 17 a 2 z 4 + 73 z 4 a -2 + 15 z 4 a -4 -7 z 4 a -6 + z 4 a -8 + 67 z 4 + 3 a 3 z 3 + 13 a z 3 + 26 z 3 a -1 + 27 z 3 a -3 + 10 z 3 a -5 - z 3 a -7 -8 a 2 z 2 -30 z 2 a -2 -7 z 2 a -4 + 2 z 2 a -6 -29 z 2 - a 3 z -3 a z -7 z a -1 -7 z a -3 -2 z a -5 + 4 a -2 + 2 a -4 + 3-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 =

2 is the signature of L11a403. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 1$	$i = 3$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{6}$
$r = -1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 0$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{11}$
$r = 1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{11}$
$r = 3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$