L11a111

(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_18,10,19,9 X_22,14,5,13 X_8,18,9,17 X_10,22,11,21 X_20,12,21,11 X_12,20,13,19 X₂₅₃₆ X_4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 4, -7, 8, -9, 5, -2, 11, -3, 6, -4, 9, -8, 7, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 5 + 2 u 5 + 3 v u 4 -3 u 4 -3 v u 3 + 3 u 3 + 3 v u 2 -3 u 2 -3 v u + 3 u + 2 v -2$ (db)
Jones polynomial	$q^{23/2}-2 q^{21/2}+4 q^{19/2}-6 q^{17/2}+9 q^{15/2}-10 q^{13/2}+9 q^{11/2}-9 q^{9/2}+6 q^{7/2}-5 q^{5/2}+2 q^{3/2}-\sqrt{q}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$- z 7 a -5 - z 7 a -7 + z 5 a -3 -4 z 5 a -5 -5 z 5 a -7 + z 5 a -9 + 4 z 3 a -3 -3 z 3 a -5 -9 z 3 a -7 + 4 z 3 a -9 + 4 z a -3 + z a -5 -9 z a -7 + 4 z a -9 + a -3 z -1 + a -5 z -1 -4 a -7 z -1 + 2 a -9 z -1$ (db)
Kauffman polynomial	$- z 10 a -6 - z 10 a -8 -2 z 9 a -5 -5 z 9 a -7 -3 z 9 a -9 -2 z 8 a -4 - z 8 a -8 -3 z 8 a -10 - z 7 a -3 + 6 z 7 a -5 + 21 z 7 a -7 + 11 z 7 a -9 -3 z 7 a -11 + 8 z 6 a -4 + 11 z 6 a -6 + 12 z 6 a -8 + 6 z 6 a -10 -3 z 6 a -12 + 5 z 5 a -3 - z 5 a -5 -35 z 5 a -7 -22 z 5 a -9 + 5 z 5 a -11 -2 z 5 a -13 -7 z 4 a -4 -18 z 4 a -6 -22 z 4 a -8 -4 z 4 a -10 + 6 z 4 a -12 - z 4 a -14 -8 z 3 a -3 -2 z 3 a -5 + 35 z 3 a -7 + 22 z 3 a -9 -4 z 3 a -11 + 3 z 3 a -13 - z 2 a -4 + 15 z 2 a -6 + 20 z 2 a -8 -4 z 2 a -10 -6 z 2 a -12 + 2 z 2 a -14 + 5 z a -3 -3 z a -5 -17 z a -7 -10 z a -9 - z a -11 + a -4 -5 a -6 -6 a -8 + a -10 + 2 a -12 - a -3 z -1 + a -5 z -1 + 4 a -7 z -1 + 2 a -9 z -1$ (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 =

5 is the signature of L11a111. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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