L11a319

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

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

-3 is the signature of L11a319. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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