L11a349

(Knotscape image)

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

Planar diagram presentation	X_12,1,13,2 X₈₄₉₃ X_14,6,15,5 X_18,8,19,7 X_20,9,21,10 X_10,11,1,12 X_6,14,7,13 X_4,18,5,17 X_22,15,11,16 X_2,19,3,20 X_16,21,17,22
Gauss code	{1, -10, 2, -8, 3, -7, 4, -2, 5, -6}, {6, -1, 7, -3, 9, -11, 8, -4, 10, -5, 11, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{(t(1)-1) (t(2)-1) \left(t(2)^2 t(1)^2-3 t(2) t(1)^2+t(1)^2-3 t(2)^2 t(1)+5 t(2) t(1)-3 t(1)+t(2)^2-3 t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db)
Jones polynomial	$q^{9/2}-4 q^{7/2}+10 q^{5/2}-18 q^{3/2}+23 \sqrt{q}-\frac{28}{\sqrt{q}}+\frac{27}{q^{3/2}}-\frac{24}{q^{5/2}}+\frac{18}{q^{7/2}}-\frac{10}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 5 z 3 + a 5 z -2 a 3 z 5 -3 a 3 z 3 + z 3 a -3 - a 3 z + z a -3 + a z 7 + 2 a z 5 -2 z 5 a -1 + a z 3 -3 z 3 a -1 - z a -1 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-3 a 2 z 10 -3 z 10 -10 a 3 z 9 -18 a z 9 -8 z 9 a -1 -13 a 4 z 8 -22 a 2 z 8 -8 z 8 a -2 -17 z 8 -9 a 5 z 7 + 5 a 3 z 7 + 26 a z 7 + 8 z 7 a -1 -4 z 7 a -3 -4 a 6 z 6 + 21 a 4 z 6 + 58 a 2 z 6 + 16 z 6 a -2 - z 6 a -4 + 50 z 6 - a 7 z 5 + 12 a 5 z 5 + 13 a 3 z 5 + 8 z 5 a -1 + 8 z 5 a -3 + 4 a 6 z 4 -17 a 4 z 4 -46 a 2 z 4 -11 z 4 a -2 + 2 z 4 a -4 -38 z 4 + a 7 z 3 -8 a 5 z 3 -12 a 3 z 3 -5 a z 3 -8 z 3 a -1 -6 z 3 a -3 - a 6 z 2 + 6 a 4 z 2 + 14 a 2 z 2 + 3 z 2 a -2 - z 2 a -4 + 11 z 2 + 2 a 5 z + 2 a 3 z + 2 z a -1 + 2 z a -3 + 1- a z -1 - a -1 z -1$ (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 =

-1 is the signature of L11a349. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -2$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = -1$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = 0$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{16}$
$r = 1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$