L11n35

(Knotscape image)

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

Visit L11n35's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_20,7,21,8 X_4,21,1,22 X_9,14,10,15 X₈₄₉₃ X_12,5,13,6 X_22,13,5,14 X_15,18,16,19 X_11,17,12,16 X_17,11,18,10 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 $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 2 u 3 + 7 v u 2 -7 u 2 -7 v u + 7 u + 2 v -2$ (db)
Jones polynomial	$-2 q^{3/2}+4 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{11}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{12}{q^{7/2}}-\frac{10}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 + 3 z 3 a 5 + 4 z a 5 + a 5 z -1 -2 z 5 a 3 -6 z 3 a 3 -8 z a 3 -3 a 3 z -1 + 4 z 3 a + 7 z a + 4 a z -1 -2 z a -1 -2 a -1 z -1$ (db)
Kauffman polynomial	$-2 a 5 z 9 -2 a 3 z 9 -5 a 6 z 8 -10 a 4 z 8 -5 a 2 z 8 -4 a 7 z 7 -5 a 5 z 7 -5 a 3 z 7 -4 a z 7 - a 8 z 6 + 12 a 6 z 6 + 26 a 4 z 6 + 12 a 2 z 6 - z 6 + 11 a 7 z 5 + 27 a 5 z 5 + 26 a 3 z 5 + 10 a z 5 + 2 a 8 z 4 -4 a 6 z 4 -17 a 4 z 4 -13 a 2 z 4 -2 z 4 -7 a 7 z 3 -22 a 5 z 3 -33 a 3 z 3 -21 a z 3 -3 z 3 a -1 - a 8 z 2 - a 2 z 2 + a 7 z + 6 a 5 z + 15 a 3 z + 15 a z + 5 z a -1 + a 6 + 3 a 4 + 3 a 2 + 2- a 5 z -1 -3 a 3 z -1 -4 a z -1 -2 a -1 z -1$ (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 L11n35. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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