L11n117

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_17,21,18,20 X_13,19,14,18 X_19,15,20,14 X_4,21,1,22 X_10,5,11,6 X_12,3,13,4 X_22,11,5,12 X_2,9,3,10 X_8,15,9,16
Gauss code	{1, -10, 8, -6}, {7, -1, 2, -11, 10, -7, 9, -8, -4, 5, 11, -2, -3, 4, -5, 3, 6, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 5 -2 v u 4 + 4 u 4 + 3 v u 3 -5 u 3 -5 v u 2 + 3 u 2 + 4 v u -2 u - v$ (db)
Jones polynomial	$2 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{7}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{10}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{5}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 3 a 7 + z a 7 - a 7 z -1 - z 5 a 5 - z 3 a 5 + 3 z a 5 + 4 a 5 z -1 -2 z 5 a 3 -7 z 3 a 3 -9 z a 3 -4 a 3 z -1 + 2 z 3 a + 3 z a + a z -1$ (db)
Kauffman polynomial	$- z 7 a 9 + 4 z 5 a 9 -5 z 3 a 9 + 2 z a 9 -3 z 8 a 8 + 13 z 6 a 8 -17 z 4 a 8 + 6 z 2 a 8 + a 8 -2 z 9 a 7 + 3 z 7 a 7 + 10 z 5 a 7 -16 z 3 a 7 + 5 z a 7 - a 7 z -1 -9 z 8 a 6 + 32 z 6 a 6 -28 z 4 a 6 + 3 z 2 a 6 + 4 a 6 -2 z 9 a 5 -4 z 7 a 5 + 30 z 5 a 5 -31 z 3 a 5 + 13 z a 5 -4 a 5 z -1 -6 z 8 a 4 + 14 z 6 a 4 -2 z 4 a 4 -10 z 2 a 4 + 7 a 4 -8 z 7 a 3 + 23 z 5 a 3 -25 z 3 a 3 + 14 z a 3 -4 a 3 z -1 -5 z 6 a 2 + 9 z 4 a 2 -10 z 2 a 2 + 4 a 2 - z 5 a -5 z 3 a + 4 z a - a z -1 -3 z 2 + 1$ (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 L11n117. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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