L11a513

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_16,5,17,6 X_14,3,15,4 X_4,15,5,16 X_22,18,13,17 X_10,20,11,19 X_18,10,19,9 X_20,12,21,11 X_12,22,7,21 X₂₇₃₈ X_6,13,1,14
Gauss code	{1, -10, 3, -4, 2, -11}, {10, -1, 7, -6, 8, -9}, {11, -3, 4, -2, 5, -7, 6, -8, 9, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 w 2 u 3 + v w 2 u 3 + 2 v 2 w u 3 - v w u 3 + 2 v 2 u 2 + v 2 w 2 u 2 -2 v w 2 u 2 + w 2 u 2 - v u 2 -3 v 2 w u 2 + 4 v w u 2 - w u 2 - v 2 u + v w 2 u -2 w 2 u + 2 v u + v 2 w u -4 v w u + 3 w u - u - v + v w -2 w + 1$ (db)
Jones polynomial	$- q 5 + 3 q 4 -5 q 3 + 9 q 2 -11 q + 13-12 q -1 + 11 q -2 -7 q -3 + 5 q -4 -2 q -5 + q -6$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z 8 -2 a 2 z 6 - z 6 a -2 + 6 z 6 + a 4 z 4 -10 a 2 z 4 -4 z 4 a -2 + 13 z 4 + 4 a 4 z 2 -16 a 2 z 2 -4 z 2 a -2 + 12 z 2 + 4 a 4 -9 a 2 + 5 + a 4 z -2 -2 a 2 z -2 + z -2$ (db)
Kauffman polynomial	$a 2 z 10 + z 10 + 2 a 3 z 9 + 5 a z 9 + 3 z 9 a -1 + 3 a 4 z 8 + 2 a 2 z 8 + 4 z 8 a -2 + 3 z 8 + 2 a 5 z 7 -2 a 3 z 7 -14 a z 7 -6 z 7 a -1 + 4 z 7 a -3 + a 6 z 6 -10 a 4 z 6 -14 a 2 z 6 -8 z 6 a -2 + 3 z 6 a -4 -14 z 6 -6 a 5 z 5 -8 a 3 z 5 + 14 a z 5 + 7 z 5 a -1 -8 z 5 a -3 + z 5 a -5 -4 a 6 z 4 + 12 a 4 z 4 + 27 a 2 z 4 + 6 z 4 a -2 -7 z 4 a -4 + 24 z 4 + 3 a 5 z 3 + 14 a 3 z 3 + 4 a z 3 -2 z 3 a -1 + 3 z 3 a -3 -2 z 3 a -5 + 4 a 6 z 2 -11 a 4 z 2 -25 a 2 z 2 - z 2 a -2 + 3 z 2 a -4 -14 z 2 -9 a 3 z -9 a z - a 6 + 5 a 4 + 11 a 2 - a -2 + 5 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 =

0 is the signature of L11a513. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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