L11a351

(Knotscape image)

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

Planar diagram presentation	X_12,1,13,2 X_18,9,19,10 X_14,6,15,5 X_6,12,7,11 X_22,15,11,16 X_20,8,21,7 X₈₃₉₄ X_16,21,17,22 X_4,18,5,17 X_10,13,1,14 X_2,19,3,20
Gauss code	{1, -11, 7, -9, 3, -4, 6, -7, 2, -10}, {4, -1, 10, -3, 5, -8, 9, -2, 11, -6, 8, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + 4 v 2 u 3 -5 v u 3 + 2 u 3 + 3 v 3 u 2 -13 v 2 u 2 + 14 v u 2 -4 u 2 -4 v 3 u + 14 v 2 u -13 v u + 3 u + 2 v 3 -5 v 2 + 4 v -1$ (db)
Jones polynomial	$-q^{7/2}+5 q^{5/2}-13 q^{3/2}+20 \sqrt{q}-\frac{27}{\sqrt{q}}+\frac{30}{q^{3/2}}-\frac{30}{q^{5/2}}+\frac{25}{q^{7/2}}-\frac{18}{q^{9/2}}+\frac{10}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 + 3 z 3 a 5 + 2 z a 5 -3 z 5 a 3 -5 z 3 a 3 -3 z a 3 + z 7 a + 3 z 5 a + 6 z 3 a + 4 z a + a z -1 - z 5 a -1 - z 3 a -1 -2 z a -1 - a -1 z -1$ (db)
Kauffman polynomial	$-4 a 4 z 10 -4 a 2 z 10 -9 a 5 z 9 -22 a 3 z 9 -13 a z 9 -8 a 6 z 8 -14 a 4 z 8 -24 a 2 z 8 -18 z 8 -4 a 7 z 7 + 13 a 5 z 7 + 36 a 3 z 7 + 6 a z 7 -13 z 7 a -1 - a 8 z 6 + 16 a 6 z 6 + 44 a 4 z 6 + 59 a 2 z 6 -5 z 6 a -2 + 27 z 6 + 8 a 7 z 5 -3 a 5 z 5 -9 a 3 z 5 + 19 a z 5 + 16 z 5 a -1 - z 5 a -3 + 2 a 8 z 4 -12 a 6 z 4 -33 a 4 z 4 -33 a 2 z 4 + 2 z 4 a -2 -12 z 4 -5 a 7 z 3 -2 a 5 z 3 -3 a 3 z 3 -13 a z 3 -7 z 3 a -1 - a 8 z 2 + 4 a 6 z 2 + 7 a 4 z 2 + 4 a 2 z 2 + 2 z 2 + a 7 z + 4 a z + 3 z a -1 + 1- a z -1 - a -1 z -1$ (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 L11a351. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -4$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -3$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = -2$	${\mathbb Z}^{16}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{14}$
$r = -1$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{16}$	${\mathbb Z}^{16}$
$r = 0$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{16}$
$r = 1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$