L11a311

(Knotscape image)

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

Planar diagram presentation	X_10,1,11,2 X_12,4,13,3 X_22,12,9,11 X_2,9,3,10 X_20,18,21,17 X_18,5,19,6 X_4,19,5,20 X_14,7,15,8 X_16,13,17,14 X_8,15,1,16 X_6,22,7,21
Gauss code	{1, -4, 2, -7, 6, -11, 8, -10}, {4, -1, 3, -2, 9, -8, 10, -9, 5, -6, 7, -5, 11, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{u^3 v^3-3 u^3 v^2+3 u^3 v-u^3-3 u^2 v^3+10 u^2 v^2-11 u^2 v+3 u^2+3 u v^3-11 u v^2+10 u v-3 u-v^3+3 v^2-3 v+1}{u^{3/2} v^{3/2}}$ (db)
Jones polynomial	$-\frac{14}{q^{9/2}}-q^{7/2}+\frac{19}{q^{7/2}}+4 q^{5/2}-\frac{23}{q^{5/2}}-9 q^{3/2}+\frac{22}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{8}{q^{11/2}}+15 \sqrt{q}-\frac{20}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 7 (- z) + 3 a 5 z 3 + 4 a 5 z + a 5 z -1 -3 a 3 z 5 -8 a 3 z 3 -8 a 3 z - a 3 z -1 + a z 7 + 4 a z 5 - z 5 a -1 + 8 a z 3 -2 z 3 a -1 + 5 a z -2 z a -1$ (db)
Kauffman polynomial	$- a 4 z 10 - a 2 z 10 -4 a 5 z 9 -9 a 3 z 9 -5 a z 9 -6 a 6 z 8 -17 a 4 z 8 -20 a 2 z 8 -9 z 8 -4 a 7 z 7 -5 a 5 z 7 -3 a 3 z 7 -10 a z 7 -8 z 7 a -1 - a 8 z 6 + 12 a 6 z 6 + 41 a 4 z 6 + 42 a 2 z 6 -4 z 6 a -2 + 10 z 6 + 10 a 7 z 5 + 30 a 5 z 5 + 42 a 3 z 5 + 35 a z 5 + 12 z 5 a -1 - z 5 a -3 + 2 a 8 z 4 -6 a 6 z 4 -27 a 4 z 4 -26 a 2 z 4 + 5 z 4 a -2 -2 z 4 -8 a 7 z 3 -29 a 5 z 3 -41 a 3 z 3 -29 a z 3 -8 z 3 a -1 + z 3 a -3 - a 8 z 2 + 4 a 4 z 2 + 5 a 2 z 2 -2 z 2 a -2 + 2 a 7 z + 10 a 5 z + 13 a 3 z + 8 a z + 3 z a -1 + a 4 - a 5 z -1 - a 3 z -1$ (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 =

-1 is the signature of L11a311. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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