L11n417

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_15,21,16,20 X_5,15,6,14 X_13,5,14,4 X_19,7,20,12 X_11,19,12,18 X_17,13,18,22 X_21,17,22,16 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -10, 2, 5, -4, -11}, {10, -1, 11, -2, -7, 6}, {-5, 4, -3, 9, -8, 7, -6, 3, -9, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 3 - v w 2 u 3 + w 2 u 3 - v 2 w u 3 + v w u 3 - v 2 u 2 - w 2 u 2 + v 2 u + w 2 u - v 2 - w 2 + v - v w + w$ (db)
Jones polynomial	$- q 7 + 2 q 6 -3 q 5 + 4 q 4 -4 q 3 + 5 q 2 -3 q + 4- q -1 + q -2$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$- z 6 a -2 - z 6 a -4 -5 z 4 a -2 -4 z 4 a -4 + z 4 a -6 + z 4 -9 z 2 a -2 -2 z 2 a -4 + 4 z 2 a -6 + 4 z 2 -10 a -2 + 4 a -4 + 2 a -6 - a -8 + 5-5 a -2 z -2 + 4 a -4 z -2 - a -6 z -2 + 2 z -2$ (db)
Kauffman polynomial	$z 9 a -1 + z 9 a -3 + 5 z 8 a -2 + 4 z 8 a -4 + z 8 -4 z 7 a -1 + z 7 a -3 + 5 z 7 a -5 -29 z 6 a -2 -19 z 6 a -4 + 3 z 6 a -6 -7 z 6 - z 5 a -1 -25 z 5 a -3 -23 z 5 a -5 + z 5 a -7 + 55 z 4 a -2 + 27 z 4 a -4 -10 z 4 a -6 + 18 z 4 + 18 z 3 a -1 + 50 z 3 a -3 + 31 z 3 a -5 - z 3 a -7 -46 z 2 a -2 -21 z 2 a -4 + 6 z 2 a -6 + 2 z 2 a -8 -21 z 2 -19 z a -1 -35 z a -3 -19 z a -5 -2 z a -7 + z a -9 + 22 a -2 + 13 a -4 - a -8 + 11 + 5 a -1 z -1 + 9 a -3 z -1 + 5 a -5 z -1 + a -7 z -1 -5 a -2 z -2 -4 a -4 z -2 - a -6 z -2 -2 z -2$ (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 =

4 is the signature of L11n417. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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