L11a313

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 5 + v u 5 - v 3 u 4 + 3 v 2 u 4 -3 v u 4 + u 4 + 2 v 3 u 3 -5 v 2 u 3 + 5 v u 3 -2 u 3 -2 v 3 u 2 + 5 v 2 u 2 -5 v u 2 + 2 u 2 + v 3 u -3 v 2 u + 3 v u - u + v 2 - v$ (db)
Jones polynomial	$-q^{11/2}+3 q^{9/2}-6 q^{7/2}+10 q^{5/2}-13 q^{3/2}+14 \sqrt{q}-\frac{16}{\sqrt{q}}+\frac{13}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 + z 7 a -1 - a 3 z 5 + 4 a z 5 + 4 z 5 a -1 - z 5 a -3 -3 a 3 z 3 + 5 a z 3 + 5 z 3 a -1 -3 z 3 a -3 -2 a 3 z + 2 a z + 2 z a -1 -2 z a -3 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -2 -2 z 10 -5 a z 9 -9 z 9 a -1 -4 z 9 a -3 -7 a 2 z 8 + z 8 a -2 -3 z 8 a -4 -3 z 8 -7 a 3 z 7 + 7 a z 7 + 30 z 7 a -1 + 15 z 7 a -3 - z 7 a -5 -5 a 4 z 6 + 13 a 2 z 6 + 13 z 6 a -2 + 12 z 6 a -4 + 19 z 6 -3 a 5 z 5 + 12 a 3 z 5 + 4 a z 5 -33 z 5 a -1 -18 z 5 a -3 + 4 z 5 a -5 - a 6 z 4 + 5 a 4 z 4 -9 a 2 z 4 -16 z 4 a -2 -14 z 4 a -4 -17 z 4 + 3 a 5 z 3 -11 a 3 z 3 -11 a z 3 + 19 z 3 a -1 + 12 z 3 a -3 -4 z 3 a -5 + a 6 z 2 - a 4 z 2 + 7 z 2 a -2 + 5 z 2 a -4 + 4 z 2 + 4 a 3 z + 4 a z -4 z a -1 -4 z a -3 + 1- a z -1 - a -1 z -1$ (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 L11a313. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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