L11n356

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,18,12,19 X_16,8,17,7 X_8,16,9,15 X_13,21,14,20 X_19,22,20,15 X_21,13,22,12 X_17,14,18,5 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {5, -4, -9, 3, -7, 6, -8, 7}, {10, -1, 4, -5, 11, -2, -3, 8, -6, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{(t(3)-1) \left(-t(1) t(2)^2+t(1) t(3) t(2)^2-t(3) t(2)^2+t(2)^2+t(1) t(2)-2 t(1) t(3) t(2)+t(3) t(2)-2 t(2)-t(1)+t(1) t(3)-t(3)+1\right)}{\sqrt{t(1)} t(2) t(3)}$ (db)
Jones polynomial	$- q -6 + 4 q -5 -6 q -4 + q 3 + 9 q -3 -3 q 2 -9 q -2 + 6 q + 10 q -1 -7$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a 2 z 6 - a 4 z 4 + 3 a 2 z 4 -2 z 4 - a 4 z 2 + 2 a 2 z 2 + z 2 a -2 -4 z 2 + a 4 -2 a 2 + a -2 + a 4 z -2 -2 a 2 z -2 + z -2$ (db)
Kauffman polynomial	$a 7 z 3 + 4 a 6 z 4 -2 a 6 z 2 + a 5 z 7 + 2 a 5 z 5 + 2 a 4 z 8 -3 a 4 z 6 + 8 a 4 z 4 -8 a 4 z 2 - a 4 z -2 + 4 a 4 + a 3 z 9 + 2 a 3 z 7 -5 a 3 z 5 + 4 a 3 z 3 -5 a 3 z + 2 a 3 z -1 + 5 a 2 z 8 -10 a 2 z 6 + z 6 a -2 + 5 a 2 z 4 -3 z 4 a -2 -6 a 2 z 2 + 3 z 2 a -2 -2 a 2 z -2 + 6 a 2 - a -2 + a z 9 + 4 a z 7 + 3 z 7 a -1 -16 a z 5 -9 z 5 a -1 + 11 a z 3 + 6 z 3 a -1 -5 a z + 2 a z -1 + 3 z 8 -6 z 6 -2 z 4 + 3 z 2 - z -2 + 2$ (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 L11n356. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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