L11a336

(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_14,3,15,4 X_22,11,9,12 X_2,15,3,16 X_4,19,5,20 X_20,5,21,6 X_18,21,19,22 X_6,13,7,14 X_16,7,17,8 X_12,17,13,18
Gauss code	{1, -5, 3, -6, 7, -9, 10, -2}, {2, -1, 4, -11, 9, -3, 5, -10, 11, -8, 6, -7, 8, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-3 v 2 u 3 + 3 v u 3 - u 3 -4 v 3 u 2 + 10 v 2 u 2 -11 v u 2 + 4 u 2 + 4 v 3 u -11 v 2 u + 10 v u -4 u - v 3 + 3 v 2 -3 v$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{13}{q^{11/2}}-\frac{20}{q^{13/2}}+\frac{23}{q^{15/2}}-\frac{23}{q^{17/2}}+\frac{21}{q^{19/2}}-\frac{16}{q^{21/2}}+\frac{10}{q^{23/2}}-\frac{5}{q^{25/2}}+\frac{1}{q^{27/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- z a 13 + 4 z 3 a 11 + 5 z a 11 -3 z 5 a 9 -6 z 3 a 9 -2 z a 9 + a 9 z -1 -3 z 5 a 7 -7 z 3 a 7 -5 z a 7 - a 7 z -1 - z 5 a 5 -2 z 3 a 5 - z a 5$ (db)
Kauffman polynomial	$- z 6 a 16 + z 4 a 16 -5 z 7 a 15 + 10 z 5 a 15 -4 z 3 a 15 - z a 15 -9 z 8 a 14 + 20 z 6 a 14 -11 z 4 a 14 + 2 z 2 a 14 -7 z 9 a 13 + 5 z 7 a 13 + 17 z 5 a 13 -14 z 3 a 13 + 2 z a 13 -2 z 10 a 12 -17 z 8 a 12 + 51 z 6 a 12 -43 z 4 a 12 + 15 z 2 a 12 -13 z 9 a 11 + 18 z 7 a 11 + 3 z 5 a 11 -7 z 3 a 11 - z a 11 -2 z 10 a 10 -15 z 8 a 10 + 40 z 6 a 10 -36 z 4 a 10 + 11 z 2 a 10 -6 z 9 a 9 + 2 z 7 a 9 + 7 z 5 a 9 -9 z 3 a 9 + 3 z a 9 - a 9 z -1 -7 z 8 a 8 + 7 z 6 a 8 - z 4 a 8 -3 z 2 a 8 + a 8 -6 z 7 a 7 + 10 z 5 a 7 -10 z 3 a 7 + 6 z a 7 - a 7 z -1 -3 z 6 a 6 + 4 z 4 a 6 - z 2 a 6 - z 5 a 5 + 2 z 3 a 5 - z a 5$ (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 L11a336. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -6$	$i = -4$
$r = -11$	${\mathbb Z}$
$r = -10$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -9$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -8$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{7}$
$r = -7$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = -6$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = -5$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = -4$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = -3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -1$	${\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$