L11a348

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{(2 t(2) t(1)-t(1)-t(2)+2) \left(t(2)^2 t(1)^2-2 t(2) t(1)^2+t(1)^2-2 t(2)^2 t(1)+3 t(2) t(1)-2 t(1)+t(2)^2-2 t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db)
Jones polynomial	$-q^{7/2}+4 q^{5/2}-9 q^{3/2}+17 \sqrt{q}-\frac{25}{\sqrt{q}}+\frac{28}{q^{3/2}}-\frac{30}{q^{5/2}}+\frac{26}{q^{7/2}}-\frac{20}{q^{9/2}}+\frac{13}{q^{11/2}}-\frac{6}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 3 z 7 + a z 7 - a 5 z 5 + a 3 z 5 + 3 a z 5 - z 5 a -1 -5 a 3 z 3 + 5 a z 3 -2 z 3 a -1 + 3 a 5 z -8 a 3 z + 4 a z - z a -1 + a 5 z -1 - a 3 z -1$ (db)
Kauffman polynomial	$-4 a 4 z 10 -4 a 2 z 10 -12 a 5 z 9 -22 a 3 z 9 -10 a z 9 -13 a 6 z 8 -20 a 4 z 8 -18 a 2 z 8 -11 z 8 -6 a 7 z 7 + 16 a 5 z 7 + 36 a 3 z 7 + 6 a z 7 -8 z 7 a -1 - a 8 z 6 + 26 a 6 z 6 + 59 a 4 z 6 + 49 a 2 z 6 -4 z 6 a -2 + 13 z 6 + 8 a 7 z 5 + 6 a 5 z 5 -2 a 3 z 5 + 11 a z 5 + 10 z 5 a -1 - z 5 a -3 -10 a 6 z 4 -33 a 4 z 4 -33 a 2 z 4 + 5 z 4 a -2 -5 z 4 -6 a 5 z 3 -17 a 3 z 3 -18 a z 3 -6 z 3 a -1 + z 3 a -3 -2 a 6 z 2 + a 4 z 2 + 3 a 2 z 2 -2 z 2 a -2 -2 z 2 -2 a 7 z + 3 a 5 z + 10 a 3 z + 7 a z + 2 z a -1 + a 4 - a 5 z -1 - a 3 z -1$ (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 L11a348. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -4$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{9}$
$r = -3$	${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = -2$	${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{15}$	${\mathbb Z}^{15}$
$r = -1$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{15}$	${\mathbb Z}^{15}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{14}$
$r = 1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$