L11a515

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 - v 2 w 2 u 3 + v w 2 u 3 + 2 v 2 w u 3 - v w u 3 + 2 v 2 u 2 + v 2 w 2 u 2 -2 v w 2 u 2 + w 2 u 2 - v u 2 -4 v 2 w u 2 + 4 v w u 2 - w u 2 - v 2 u + v w 2 u -2 w 2 u + 2 v u + v 2 w u -4 v w u + 4 w u - u + w 2 - v + v w -2 w + 1$ (db)
Jones polynomial	$q 7 -3 q 6 + 6 q 5 -9 q 4 + 13 q 3 -13 q 2 + 14 q -11 + 9 q -1 -5 q -2 + 3 q -3 - q -4$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$- z 8 a -2 -6 z 6 a -2 + z 6 a -4 + 2 z 6 - a 2 z 4 -14 z 4 a -2 + 4 z 4 a -4 + 9 z 4 -3 a 2 z 2 -16 z 2 a -2 + 5 z 2 a -4 + 12 z 2 - a 2 -8 a -2 + 3 a -4 + 6-2 a -2 z -2 + a -4 z -2 + z -2$ (db)
Kauffman polynomial	$z 10 a -2 + z 10 + 3 a z 9 + 7 z 9 a -1 + 4 z 9 a -3 + 3 a 2 z 8 + 9 z 8 a -2 + 6 z 8 a -4 + 6 z 8 + a 3 z 7 -9 a z 7 -21 z 7 a -1 -5 z 7 a -3 + 6 z 7 a -5 -13 a 2 z 6 -40 z 6 a -2 -11 z 6 a -4 + 5 z 6 a -6 -37 z 6 -4 a 3 z 5 + 2 a z 5 + 14 z 5 a -1 -3 z 5 a -3 -8 z 5 a -5 + 3 z 5 a -7 + 17 a 2 z 4 + 56 z 4 a -2 + 10 z 4 a -4 -6 z 4 a -6 + z 4 a -8 + 56 z 4 + 4 a 3 z 3 + 7 a z 3 + 4 z 3 a -1 + 8 z 3 a -3 + 4 z 3 a -5 -3 z 3 a -7 -9 a 2 z 2 -37 z 2 a -2 -6 z 2 a -4 + 3 z 2 a -6 - z 2 a -8 -36 z 2 - a 3 z -3 a z -8 z a -1 -6 z a -3 + 2 a 2 + 12 a -2 + 4 a -4 - a -6 + 10 + 2 a -1 z -1 + 2 a -3 z -1 -2 a -2 z -2 - a -4 z -2 - z -2$ (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 =

2 is the signature of L11a515. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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