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 $u$ , $v$ , $w$ , ...)	$-\frac{3 u^3 v^2-3 u^3 v+u^3+4 u^2 v^3-10 u^2 v^2+11 u^2 v-4 u^2-4 u v^3+11 u v^2-10 u v+4 u+v^3-3 v^2+3 v}{u^{3/2} v^{3/2}}$ (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	$a 16 z 6 - a 16 z 4 + 5 a 15 z 7 -10 a 15 z 5 + 4 a 15 z 3 + a 15 z + 9 a 14 z 8 -20 a 14 z 6 + 11 a 14 z 4 -2 a 14 z 2 + 7 a 13 z 9 -5 a 13 z 7 -17 a 13 z 5 + 14 a 13 z 3 -2 a 13 z + 2 a 12 z 10 + 17 a 12 z 8 -51 a 12 z 6 + 43 a 12 z 4 -15 a 12 z 2 + 13 a 11 z 9 -18 a 11 z 7 -3 a 11 z 5 + 7 a 11 z 3 + a 11 z + 2 a 10 z 10 + 15 a 10 z 8 -40 a 10 z 6 + 36 a 10 z 4 -11 a 10 z 2 + 6 a 9 z 9 -2 a 9 z 7 -7 a 9 z 5 + 9 a 9 z 3 -3 a 9 z + a 9 z -1 + 7 a 8 z 8 -7 a 8 z 6 + a 8 z 4 + 3 a 8 z 2 - a 8 + 6 a 7 z 7 -10 a 7 z 5 + 10 a 7 z 3 -6 a 7 z + a 7 z -1 + 3 a 6 z 6 -4 a 6 z 4 + a 6 z 2 + a 5 z 5 -2 a 5 z 3 + a 5 z$ (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.

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