L11n217

(Knotscape image)

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

Planar diagram presentation	X_10,1,11,2 X_8,9,1,10 X_3,12,4,13 X_15,22,16,9 X_17,3,18,2 X_21,4,22,5 X_5,15,6,14 X_13,21,14,20 X_11,16,12,17 X_19,7,20,6 X_7,19,8,18
Gauss code	{1, 5, -3, 6, -7, 10, -11, -2}, {2, -1, -9, 3, -8, 7, -4, 9, -5, 11, -10, 8, -6, 4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 + v u 3 -2 v 3 u 2 + 6 v 2 u 2 -6 v u 2 + 2 u 2 + 2 v 3 u -6 v 2 u + 6 v u -2 u + v 2 - v$ (db)
Jones polynomial	$2 q^{5/2}-6 q^{3/2}+8 \sqrt{q}-\frac{12}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{10}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 3 a 5 + z a 5 - z 5 a 3 - z 3 a 3 -2 z 5 a -5 z 3 a -3 z a + a z -1 + 2 z 3 a -1 + 2 z a -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 a 3 z 9 -2 a z 9 -5 a 4 z 8 -8 a 2 z 8 -3 z 8 -5 a 5 z 7 -2 a 3 z 7 + 2 a z 7 - z 7 a -1 -3 a 6 z 6 + 11 a 4 z 6 + 19 a 2 z 6 + 5 z 6 - a 7 z 5 + 11 a 5 z 5 + 8 a 3 z 5 -9 a z 5 -5 z 5 a -1 + 6 a 6 z 4 -11 a 4 z 4 -23 a 2 z 4 -3 z 4 a -2 -9 z 4 + 2 a 7 z 3 -8 a 5 z 3 -6 a 3 z 3 + 15 a z 3 + 11 z 3 a -1 -2 a 6 z 2 + 4 a 4 z 2 + 11 a 2 z 2 + 3 z 2 a -2 + 8 z 2 + 2 a 5 z -6 a z -4 z a -1 + 1- a z -1 - a -1 z -1$ (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 L11n217. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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