L11n314

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_3,12,4,13 X_7,17,8,16 X_9,20,10,21 X_11,18,12,19 X_19,22,20,11 X_15,9,16,8 X_21,10,22,5 X_17,14,18,15 X₂₅₃₆ X_13,4,14,1
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 7, -4, 8}, {-5, 2, -11, 9, -7, 3, -9, 5, -6, 4, -8, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 2 -2 v 2 w u 2 + 3 v w u 2 - w u 2 + u 2 - v 2 u + 3 v u + 3 v 2 w u -3 v w u + w u -3 u + v 2 -3 v - v 2 w + v w + 2$ (db)
Jones polynomial	$1-2 q -1 + 6 q -2 -7 q -3 + 10 q -4 -10 q -5 + 10 q -6 -7 q -7 + 5 q -8 -2 q -9$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- a 10 + z 4 a 8 + 3 z 2 a 8 + 2 a 8 - z 6 a 6 -3 z 4 a 6 -2 z 2 a 6 + a 6 z -2 - z 6 a 4 -3 z 4 a 4 -3 z 2 a 4 -2 a 4 z -2 -4 a 4 + z 4 a 2 + 3 z 2 a 2 + a 2 z -2 + 3 a 2$ (db)
Kauffman polynomial	$3 z 3 a 11 -2 z a 11 + z 6 a 10 + 4 z 4 a 10 -5 z 2 a 10 + 2 a 10 + 3 z 7 a 9 -4 z 5 a 9 + 8 z 3 a 9 -4 z a 9 + 3 z 8 a 8 -5 z 6 a 8 + 11 z 4 a 8 -12 z 2 a 8 + 4 a 8 + z 9 a 7 + 4 z 7 a 7 -9 z 5 a 7 + 4 z 3 a 7 + 5 z 8 a 6 -8 z 6 a 6 + z 2 a 6 + a 6 z -2 -2 a 6 + z 9 a 5 + 3 z 7 a 5 -10 z 5 a 5 + 6 z a 5 -2 a 5 z -1 + 2 z 8 a 4 - z 6 a 4 -11 z 4 a 4 + 14 z 2 a 4 + 2 a 4 z -2 -7 a 4 + 2 z 7 a 3 -5 z 5 a 3 + z 3 a 3 + 4 z a 3 -2 a 3 z -1 + z 6 a 2 -4 z 4 a 2 + 6 z 2 a 2 + a 2 z -2 -4 a 2$ (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 L11n314. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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