L11n261

(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_18,12,19,11 X_22,20,9,19 X_20,16,21,15 X_16,22,17,21 X_12,18,13,17 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 $u$ , $v$ , $w$ , ...)	$v u 3 - v w u 3 + w u 3 + u 3 -2 v u 2 -2 w u 2 - u 2 + 2 v u + v w u + 2 w u - v - v w - w + 1$ (db)
Jones polynomial	$- q 10 + q 9 - q 8 - q 7 + 3 q 6 -3 q 5 + 5 q 4 -3 q 3 + 5 q 2 -2 q + 1$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$- z 6 a -4 + z 4 a -2 -4 z 4 a -4 + 3 z 2 a -2 -4 z 2 a -4 -2 z 2 a -6 + 2 z 2 a -8 + 3 a -2 -2 a -4 -5 a -6 + 5 a -8 - a -10 + a -2 z -2 - a -4 z -2 -2 a -6 z -2 + 3 a -8 z -2 - a -10 z -2$ (db)
Kauffman polynomial	$z 8 a -4 + z 8 a -6 + z 8 a -8 + z 8 a -10 + 2 z 7 a -3 + 4 z 7 a -5 + 3 z 7 a -7 + 2 z 7 a -9 + z 7 a -11 + z 6 a -2 + z 6 a -4 -2 z 6 a -6 -8 z 6 a -8 -6 z 6 a -10 -6 z 5 a -3 -15 z 5 a -5 -20 z 5 a -7 -17 z 5 a -9 -6 z 5 a -11 -4 z 4 a -2 -14 z 4 a -4 -3 z 4 a -6 + 15 z 4 a -8 + 8 z 4 a -10 + z 3 a -3 + 16 z 3 a -5 + 42 z 3 a -7 + 37 z 3 a -9 + 10 z 3 a -11 + 6 z 2 a -2 + 12 z 2 a -4 + 2 z 2 a -6 -7 z 2 a -8 -3 z 2 a -10 + 4 z a -3 -10 z a -5 -34 z a -7 -27 z a -9 -7 z a -11 -4 a -2 -3 a -4 + 4 a -6 + 5 a -8 + a -10 -2 a -3 z -1 + 2 a -5 z -1 + 10 a -7 z -1 + 8 a -9 z -1 + 2 a -11 z -1 + a -2 z -2 + a -4 z -2 -2 a -6 z -2 -3 a -8 z -2 - a -10 z -2$ (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 L11n261. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 1$	$i = 3$	$i = 5$
$r = -2$		${\mathbb Z}$
$r = -1$		${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 0$		${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}^{4}$
$r = 1$		${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 2$		${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 3$	${\mathbb Z}$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 5$	${\mathbb Z}$	${\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$
$r = 8$	${\mathbb Z}$
$r = 9$	${\mathbb Z}_2$	${\mathbb Z}$