L11n415

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_5,15,6,14 X_10,3,11,4 X_13,5,14,4 X₂₇₃₈ X_6,9,1,10 X_18,12,19,11 X_12,18,7,17 X_20,16,21,15 X_22,20,13,19 X_16,22,17,21
Gauss code	{1, -5, 3, 4, -2, -6}, {5, -1, 6, -3, 7, -8}, {-4, 2, 9, -11, 8, -7, 10, -9, 11, -10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 3 - v w u 3 + w u 3 + v 2 u 2 -2 v w 2 u 2 + w 2 u 2 -2 v u 2 -2 v 2 w u 2 + 4 v w u 2 -2 w u 2 - v 2 u + 2 v w 2 u - w 2 u + 2 v u + 2 v 2 w u -4 v w u + 2 w u - v w 2 - v 2 w + v w$ (db)
Jones polynomial	$- q 8 + 3 q 7 -6 q 6 + 9 q 5 -11 q 4 + 12 q 3 -10 q 2 + 9 q -4 + 3 q -1$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-3 z 4 a -2 -2 z 4 a -4 -8 z 2 a -2 - z 2 a -4 + 3 z 2 a -6 + 3 z 2 -10 a -2 + 4 a -4 + 2 a -6 - a -8 + 5-5 a -2 z -2 + 4 a -4 z -2 - a -6 z -2 + 2 z -2$ (db)
Kauffman polynomial	$z 9 a -3 + z 9 a -5 + 4 z 8 a -2 + 7 z 8 a -4 + 3 z 8 a -6 + 3 z 7 a -1 + 7 z 7 a -3 + 8 z 7 a -5 + 4 z 7 a -7 -14 z 6 a -2 -17 z 6 a -4 + 3 z 6 a -8 -9 z 5 a -1 -28 z 5 a -3 -26 z 5 a -5 -6 z 5 a -7 + z 5 a -9 + 32 z 4 a -2 + 23 z 4 a -4 -9 z 4 a -6 -6 z 4 a -8 + 6 z 4 + 18 z 3 a -1 + 48 z 3 a -3 + 33 z 3 a -5 + z 3 a -7 -2 z 3 a -9 -38 z 2 a -2 -19 z 2 a -4 + 6 z 2 a -6 + 3 z 2 a -8 -16 z 2 -19 z a -1 -35 z a -3 -19 z a -5 -2 z a -7 + z a -9 + 22 a -2 + 13 a -4 - a -8 + 11 + 5 a -1 z -1 + 9 a -3 z -1 + 5 a -5 z -1 + a -7 z -1 -5 a -2 z -2 -4 a -4 z -2 - a -6 z -2 -2 z -2$ (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 L11n415. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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