L11a510

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 - v 2 w 2 u 3 + v w 2 u 3 + 2 v 2 w u 3 - v w u 3 + 2 v 2 u 2 + v 2 w 2 u 2 -2 v w 2 u 2 + w 2 u 2 - v u 2 -2 v 2 w u 2 + 2 v w u 2 - w u 2 - v 2 u + v w 2 u -2 w 2 u + 2 v u + v 2 w u -2 v w u + 2 w u - u + w 2 - v + v w -2 w + 1$ (db)
Jones polynomial	$- q 9 + 3 q 8 -5 q 7 + 8 q 6 -10 q 5 + 11 q 4 -10 q 3 + 10 q 2 -6 q + 5-2 q -1 + q -2$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z 8 a -4 -2 z 6 a -2 + 6 z 6 a -4 - z 6 a -6 -10 z 4 a -2 + 13 z 4 a -4 -4 z 4 a -6 + z 4 -15 z 2 a -2 + 13 z 2 a -4 -4 z 2 a -6 + 4 z 2 -8 a -2 + 5 a -4 - a -6 + 4-2 a -2 z -2 + a -4 z -2 + z -2$ (db)
Kauffman polynomial	$z 10 a -2 + z 10 a -4 + 2 z 9 a -1 + 6 z 9 a -3 + 4 z 9 a -5 + z 8 a -2 + 7 z 8 a -4 + 7 z 8 a -6 + z 8 -10 z 7 a -1 -24 z 7 a -3 -6 z 7 a -5 + 8 z 7 a -7 -24 z 6 a -2 -42 z 6 a -4 -17 z 6 a -6 + 7 z 6 a -8 -6 z 6 + 14 z 5 a -1 + 22 z 5 a -3 -13 z 5 a -5 -16 z 5 a -7 + 5 z 5 a -9 + 51 z 4 a -2 + 59 z 4 a -4 + 9 z 4 a -6 -9 z 4 a -8 + 3 z 4 a -10 + 13 z 4 -2 z 3 a -1 + 4 z 3 a -3 + 17 z 3 a -5 + 7 z 3 a -7 -3 z 3 a -9 + z 3 a -11 -38 z 2 a -2 -32 z 2 a -4 -4 z 2 a -6 + 2 z 2 a -8 - z 2 a -10 -13 z 2 -6 z a -1 -8 z a -3 -3 z a -5 - z a -7 + 12 a -2 + 8 a -4 + a -6 + 6 + 2 a -1 z -1 + 2 a -3 z -1 -2 a -2 z -2 - a -4 z -2 - z -2$ (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 =

4 is the signature of L11a510. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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