L11a117

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,4,15,3 X_16,8,17,7 X_22,14,5,13 X_8,22,9,21 X_20,10,21,9 X_18,12,19,11 X_12,18,13,17 X_10,20,11,19 X₂₅₃₆ X_4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -5, 6, -9, 7, -8, 4, -2, 11, -3, 8, -7, 9, -6, 5, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-3 v u 3 + 3 u 3 + 5 v u 2 -5 u 2 -5 v u + 5 u + 3 v -3$ (db)
Jones polynomial	$q^{21/2}-2 q^{19/2}+4 q^{17/2}-6 q^{15/2}+8 q^{13/2}-10 q^{11/2}+10 q^{9/2}-9 q^{7/2}+6 q^{5/2}-5 q^{3/2}+2 \sqrt{q}-\frac{1}{\sqrt{q}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$- z 5 a -3 - z 5 a -5 - z 5 a -7 + z 3 a -1 -2 z 3 a -3 - z 3 a -5 -3 z 3 a -7 + z 3 a -9 + 2 z a -1 - z a -3 + z a -5 -4 z a -7 + 2 z a -9 + a -1 z -1 - a -3 z -1 + a -5 z -1 -2 a -7 z -1 + a -9 z -1$ (db)
Kauffman polynomial	$- z 10 a -6 - z 10 a -8 -2 z 9 a -5 -5 z 9 a -7 -3 z 9 a -9 -2 z 8 a -4 - z 8 a -8 -3 z 8 a -10 -2 z 7 a -3 + 5 z 7 a -5 + 22 z 7 a -7 + 13 z 7 a -9 -2 z 7 a -11 -2 z 6 a -2 + 2 z 6 a -4 + 9 z 6 a -6 + 18 z 6 a -8 + 12 z 6 a -10 - z 6 a -12 - z 5 a -1 + z 5 a -3 -8 z 5 a -5 -38 z 5 a -7 -21 z 5 a -9 + 7 z 5 a -11 + 4 z 4 a -2 + z 4 a -4 -23 z 4 a -6 -39 z 4 a -8 -15 z 4 a -10 + 4 z 4 a -12 + 3 z 3 a -1 + 6 z 3 a -3 + 7 z 3 a -5 + 25 z 3 a -7 + 17 z 3 a -9 -4 z 3 a -11 + 15 z 2 a -6 + 29 z 2 a -8 + 11 z 2 a -10 -3 z 2 a -12 -3 z a -1 -3 z a -3 -5 z a -5 -10 z a -7 -5 z a -9 - a -2 -4 a -6 -7 a -8 -3 a -10 + a -1 z -1 + a -3 z -1 + a -5 z -1 + 2 a -7 z -1 + a -9 z -1$ (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 =

3 is the signature of L11a117. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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