L11n259

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_7,15,8,14 X_13,5,14,8 X_11,18,12,19 X_22,20,9,19 X_20,16,21,15 X_16,22,17,21 X_17,12,18,13 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, -5, 9, -4, 3, 7, -8, -9, 5, 6, -7, 8, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 3 - v w u 3 + w u 3 - u 3 -2 v u 2 + 2 v w u 2 -2 w u 2 + u 2 + 2 v u - v w u + 2 w u -2 u - v + v w - w + 1$ (db)
Jones polynomial	$- q 6 + 4 q 5 -6 q 4 + 8 q 3 -7 q 2 + 8 q -5 + 4 q -1 - q -2$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + 3 z 4 a -2 - z 4 a -4 - z 4 + z 2 a -2 - z 2 a -4 - z 2 -3 a -2 + a -4 + 2-2 a -2 z -2 + a -4 z -2 + z -2$ (db)
Kauffman polynomial	$z 8 a -2 + z 8 a -4 + 4 z 7 a -1 + 5 z 7 a -3 + z 7 a -5 + 6 z 6 a -2 + 2 z 6 a -4 + 4 z 6 + a z 5 -7 z 5 a -1 -6 z 5 a -3 + 2 z 5 a -5 -14 z 4 a -2 - z 4 a -4 + 4 z 4 a -6 -9 z 4 - a z 3 + z 3 a -1 + z 3 a -3 + z 3 a -7 + z 2 a -2 -3 z 2 a -4 - z 2 a -6 + 3 z 2 -3 z a -1 -3 z a -3 + 5 a -2 + 3 a -4 + 3 + 2 a -1 z -1 + 2 a -3 z -1 -2 a -2 z -2 - a -4 z -2 - z -2$ (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 L11n259. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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