L11a516

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 3 - v w 2 u 3 + w 2 u 3 - v u 3 - v 2 w u 3 + 2 v w u 3 - w u 3 -3 v 2 u 2 - v 2 w 2 u 2 + 3 v w 2 u 2 -3 w 2 u 2 + 3 v u 2 + 4 v 2 w u 2 -6 v w u 2 + 4 w u 2 - u 2 + 3 v 2 u + v 2 w 2 u -3 v w 2 u + 3 w 2 u -3 v u -4 v 2 w u + 6 v w u -4 w u + u - v 2 + v w 2 - w 2 + v + v 2 w -2 v w + w$ (db)
Jones polynomial	$q 4 -4 q 3 + 9 q 2 -15 q + 21-22 q -1 + 24 q -2 -19 q -3 + 15 q -4 -9 q -5 + 4 q -6 - q -7$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$- z 2 a 6 - a 6 + 3 z 4 a 4 + 6 z 2 a 4 + a 4 z -2 + 3 a 4 -2 z 6 a 2 -6 z 4 a 2 -7 z 2 a 2 -2 a 2 z -2 -4 a 2 - z 6 - z 4 + z 2 + z -2 + 2 + z 4 a -2 + z 2 a -2$ (db)
Kauffman polynomial	$3 a 4 z 10 + 3 a 2 z 10 + 6 a 5 z 9 + 17 a 3 z 9 + 11 a z 9 + 4 a 6 z 8 + 7 a 4 z 8 + 19 a 2 z 8 + 16 z 8 + a 7 z 7 -16 a 5 z 7 -44 a 3 z 7 -13 a z 7 + 14 z 7 a -1 -13 a 6 z 6 -48 a 4 z 6 -73 a 2 z 6 + 9 z 6 a -2 -29 z 6 -3 a 7 z 5 + 7 a 5 z 5 + 24 a 3 z 5 -8 a z 5 -18 z 5 a -1 + 4 z 5 a -3 + 14 a 6 z 4 + 60 a 4 z 4 + 74 a 2 z 4 -7 z 4 a -2 + z 4 a -4 + 20 z 4 + 3 a 7 z 3 + 6 a 5 z 3 + 3 a 3 z 3 + 8 a z 3 + 7 z 3 a -1 - z 3 a -3 -7 a 6 z 2 -29 a 4 z 2 -33 a 2 z 2 + 2 z 2 a -2 -9 z 2 - a 7 z -3 a 5 z -4 a 3 z -2 a z + 2 a 6 + 7 a 4 + 7 a 2 + 3 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 L11a516. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -4$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -3$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = -2$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{12}$
$r = -1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 0$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{12}$
$r = 1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$