L11n34

(Knotscape image)

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

Visit L11n34's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_20,7,21,8 X_4,21,1,22 X_14,10,15,9 X₈₄₉₃ X_5,13,6,12 X_13,5,14,22 X_18,16,19,15 X_16,11,17,12 X_10,17,11,18 X_2,20,3,19
Gauss code	{1, -11, 5, -3}, {-6, -1, 2, -5, 4, -10, 9, 6, -7, -4, 8, -9, 10, -8, 11, -2, 3, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 4 v u 4 -4 u 4 -6 v u 3 + 6 u 3 + 6 v u 2 -6 u 2 -4 v u + 4 u + v -1$ (db)
Jones polynomial	$-3 q^{9/2}+7 q^{7/2}-11 q^{5/2}+14 q^{3/2}-15 \sqrt{q}+\frac{14}{\sqrt{q}}-\frac{12}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 2 a z 5 -4 z 5 a -1 + z 5 a -3 - a 3 z 3 + 5 a z 3 -6 z 3 a -1 + 3 z 3 a -3 - a 3 z + 2 a z -4 z a -1 + 4 z a -3 - z a -5 + a 3 z -1 - a z -1 - a -1 z -1 + 2 a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$-2 a z 9 -2 z 9 a -1 -5 a 2 z 8 -7 z 8 a -2 -12 z 8 -4 a 3 z 7 -9 a z 7 -13 z 7 a -1 -8 z 7 a -3 - a 4 z 6 + 10 a 2 z 6 + 10 z 6 a -2 -3 z 6 a -4 + 24 z 6 + 11 a 3 z 5 + 35 a z 5 + 40 z 5 a -1 + 16 z 5 a -3 + 2 a 4 z 4 -2 a 2 z 4 -8 z 4 a -2 -3 z 4 a -4 -9 z 4 -9 a 3 z 3 -27 a z 3 -36 z 3 a -1 -24 z 3 a -3 -6 z 3 a -5 - a 4 z 2 - a 2 z 2 + 6 z 2 a -2 + 3 z 2 a -4 + 3 z 2 + a 3 z + 5 a z + 14 z a -1 + 15 z a -3 + 5 z a -5 - a 2 -3 a -2 - a -4 -2 + a 3 z -1 + a z -1 - a -1 z -1 -2 a -3 z -1 - a -5 z -1$ (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 =

1 is the signature of L11n34. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 0$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{8}$
$r = 1$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 4$	${\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$