L11n367

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,8,13,7 X_13,20,14,21 X_19,9,20,8 X_9,19,10,18 X_15,22,16,17 X_17,16,18,5 X_21,14,22,15 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {-8, 6, -5, 4, -9, 7}, {10, -1, 3, 5, -6, -2, 11, -3, -4, 9, -7, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 2 - v u 2 - v 2 w u 2 + v w u 2 - v + v w - w + 1$ (db)
Jones polynomial	$- q 4 + 2 q 3 -2 q 2 + 3 q -2 + 3 q -1 - q -2 + 2 q -3$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z 6 - a 2 z 4 - z 4 a -2 + 5 z 4 -4 a 2 z 2 -3 z 2 a -2 + 7 z 2 + a 4 -4 a 2 - a -2 + 4 + a 4 z -2 -2 a 2 z -2 + z -2$ (db)
Kauffman polynomial	$a z 9 + z 9 a -1 + a 2 z 8 + 2 z 8 a -2 + 3 z 8 -5 a z 7 -4 z 7 a -1 + z 7 a -3 -6 a 2 z 6 -11 z 6 a -2 -17 z 6 + 6 a z 5 + z 5 a -1 -5 z 5 a -3 + 12 a 2 z 4 + 17 z 4 a -2 + 29 z 4 - a z 3 + 5 z 3 a -1 + 6 z 3 a -3 - a 4 z 2 -13 a 2 z 2 -9 z 2 a -2 -21 z 2 -2 a 3 z -4 a z -3 z a -1 - z a -3 + 3 a 4 + 7 a 2 + 2 a -2 + 7 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 L11n367. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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