L11a133

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_10,5,11,6 X_14,3,15,4 X_18,11,19,12 X_22,19,5,20 X_20,14,21,13 X_12,22,13,21 X_2,9,3,10 X_8,15,9,16
Gauss code	{1, -10, 5, -3}, {4, -1, 2, -11, 10, -4, 6, -9, 8, -5, 11, -2, 3, -6, 7, -8, 9, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 5 -4 v u 4 + 6 u 4 + 14 v u 3 -14 u 3 -14 v u 2 + 14 u 2 + 6 v u -4 u - v$ (db)
Jones polynomial	$-\sqrt{q}+\frac{4}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{15}{q^{5/2}}-\frac{22}{q^{7/2}}+\frac{25}{q^{9/2}}-\frac{25}{q^{11/2}}+\frac{22}{q^{13/2}}-\frac{16}{q^{15/2}}+\frac{10}{q^{17/2}}-\frac{5}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z 3 a 9 + z a 9 + a 9 z -1 + z 5 a 7 -2 z 3 a 7 -5 z a 7 -3 a 7 z -1 + 3 z 5 a 5 + 6 z 3 a 5 + 7 z a 5 + 4 a 5 z -1 + z 5 a 3 -2 z 3 a 3 -5 z a 3 -2 a 3 z -1 - z 3 a$ (db)
Kauffman polynomial	$- z 6 a 12 + z 4 a 12 -5 z 7 a 11 + 10 z 5 a 11 -4 z 3 a 11 - z a 11 -9 z 8 a 10 + 20 z 6 a 10 -12 z 4 a 10 + z 2 a 10 + a 10 -7 z 9 a 9 + 4 z 7 a 9 + 17 z 5 a 9 -14 z 3 a 9 + 3 z a 9 - a 9 z -1 -2 z 10 a 8 -19 z 8 a 8 + 52 z 6 a 8 -36 z 4 a 8 + 5 z 2 a 8 + 3 a 8 -14 z 9 a 7 + 13 z 7 a 7 + 24 z 5 a 7 -33 z 3 a 7 + 13 z a 7 -3 a 7 z -1 -2 z 10 a 6 -21 z 8 a 6 + 52 z 6 a 6 -38 z 4 a 6 + 5 z 2 a 6 + 3 a 6 -7 z 9 a 5 -5 z 7 a 5 + 33 z 5 a 5 -36 z 3 a 5 + 16 z a 5 -4 a 5 z -1 -11 z 8 a 4 + 17 z 6 a 4 -11 z 4 a 4 + z 2 a 4 + 2 a 4 -9 z 7 a 3 + 15 z 5 a 3 -12 z 3 a 3 + 7 z a 3 -2 a 3 z -1 -4 z 6 a 2 + 4 z 4 a 2 - z 5 a + z 3 a$ (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 L11a133. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$
$r = -9$	${\mathbb Z}$
$r = -8$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -6$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -5$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -4$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{13}$
$r = -3$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = -2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = -1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 0$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{7}$
$r = 1$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 2$	${\mathbb Z}_2$	${\mathbb Z}$