L11n56

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n56 at Knotilus!
	[edit L11n56 Quick Notes]

[edit L11n56 Further Notes and Views]

Link Presentations

[edit Notes on L11n56's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,17,12,16 X_14,7,15,8 X_8,15,9,16 X_13,21,14,20 X_17,5,18,22 X_21,19,22,18 X_19,13,20,12 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, -3, 9, -6, -4, 5, 3, -7, 8, -9, 6, -8, 7}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(1)t(2)^{3}-t(2)^{3}-4t(1)t(2)^{2}+4t(2)^{2}+4t(1)t(2)-4t(2)-t(1)+2}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {1}{q^{9/2}}}-3q^{7/2}+{\frac {2}{q^{7/2}}}+5q^{5/2}-{\frac {5}{q^{5/2}}}-6q^{3/2}+{\frac {6}{q^{3/2}}}+7{\sqrt {q}}-{\frac {8}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{3}+z^{3}a^{-3}+2a^{3}z+za^{-3}+2a^{3}z^{-1}-az^{5}-z^{5}a^{-1}-3az^{3}-2z^{3}a^{-1}-5az-3az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-a^{2}z^{8}-3z^{8}a^{-2}-4z^{8}+2az^{7}-z^{7}a^{-1}-3z^{7}a^{-3}+2a^{2}z^{6}+9z^{6}a^{-2}-z^{6}a^{-4}+12z^{6}-4a^{3}z^{5}-5az^{5}+9z^{5}a^{-1}+10z^{5}a^{-3}-2a^{4}z^{4}-7a^{2}z^{4}-5z^{4}a^{-2}+3z^{4}a^{-4}-13z^{4}-a^{5}z^{3}+6a^{3}z^{3}+9az^{3}-5z^{3}a^{-1}-7z^{3}a^{-3}+a^{4}z^{2}+7a^{2}z^{2}+z^{2}a^{-2}-2z^{2}a^{-4}+9z^{2}+a^{5}z-6a^{3}z-9az-za^{-1}+za^{-3}-3a^{2}-a^{-2}-3+2a^{3}z^{-1}+3az^{-1}+a^{-1}z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

\		r
	\
j		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

2

6

3

1

-2

4

3

2

1

2

4

3

-1

0

4

3

1

-2

3

5

2

-4

2

3

-1

-6

3

-8

1

2

-1

-10

1

Integral Khovanov Homology

(db, data source)

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