L11n4

(Knotscape image)

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

Visit L11n4's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_9,14,10,15 X₃₈₄₉ X_5,11,6,10 X_11,19,12,18 X_13,20,14,21 X_19,5,20,22 X_21,12,22,13 X_15,2,16,3
Gauss code	{1, 11, -5, -3}, {-6, -1, 2, 5, -4, 6, -7, 10, -8, 4, -11, -2, 3, 7, -9, 8, -10, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 4 v u 4 -6 v u 3 + 2 u 3 + 2 v u 2 -6 u 2 + 4 u -1$ (db)
Jones polynomial	$q^{3/2}-3 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{9}{q^{7/2}}+\frac{8}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{2}{q^{15/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a 9 z -1 - z 3 a 7 -4 z a 7 -3 a 7 z -1 + 2 z 5 a 5 + 8 z 3 a 5 + 9 z a 5 + 4 a 5 z -1 - z 7 a 3 -5 z 5 a 3 -9 z 3 a 3 -8 z a 3 -2 a 3 z -1 + z 5 a + 3 z 3 a + z a$ (db)
Kauffman polynomial	$-3 z 3 a 9 + 4 z a 9 - a 9 z -1 - z 6 a 8 -2 z 4 a 8 + 4 z 2 a 8 - a 8 -3 z 7 a 7 + 9 z 5 a 7 -19 z 3 a 7 + 15 z a 7 -3 a 7 z -1 -3 z 8 a 6 + 9 z 6 a 6 -16 z 4 a 6 + 11 z 2 a 6 -3 a 6 - z 9 a 5 -4 z 7 a 5 + 23 z 5 a 5 -39 z 3 a 5 + 24 z a 5 -4 a 5 z -1 -6 z 8 a 4 + 20 z 6 a 4 -22 z 4 a 4 + 9 z 2 a 4 -2 a 4 - z 9 a 3 -4 z 7 a 3 + 25 z 5 a 3 -33 z 3 a 3 + 16 z a 3 -2 a 3 z -1 -3 z 8 a 2 + 9 z 6 a 2 -5 z 4 a 2 + z 2 a 2 - a 2 -3 z 7 a + 11 z 5 a -10 z 3 a + 3 z a - z 6 + 3 z 4 - z 2$ (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 =

-3 is the signature of L11n4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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