L11n122

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n122 at Knotilus!
	[edit L11n122 Quick Notes]

[edit L11n122 Further Notes and Views]

Link Presentations

[edit Notes on L11n122's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_14,3,15,4 X_16,10,17,9 X_11,21,12,20 X_21,9,22,8 X_7,19,8,18 X_19,13,20,12 X_10,16,11,15 X_22,17,5,18 X₂₅₃₆ X_4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, -6, 5, 3, -8, -4, 7, 11, -2, 8, -3, 9, 6, -7, 4, -5, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {-2uv^{2}+2uv+u+v^{3}+2v^{2}-2v}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-q^{11/2}+2q^{9/2}-2q^{7/2}+3q^{5/2}-2q^{3/2}+{\sqrt {q}}-{\frac {2}{\sqrt {q}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$za^{3}+2a^{3}z^{-1}-z^{3}a-5za-3az^{-1}+z^{3}a^{-1}+2za^{-1}+a^{-1}z^{-1}+z^{3}a^{-3}+za^{-3}-za^{-5}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-3z^{8}a^{-2}-2z^{8}a^{-4}-z^{8}+5z^{7}a^{-1}+4z^{7}a^{-3}-z^{7}a^{-5}+17z^{6}a^{-2}+11z^{6}a^{-4}+6z^{6}-a^{3}z^{5}-5z^{5}a^{-1}-z^{5}a^{-3}+5z^{5}a^{-5}-26z^{4}a^{-2}-17z^{4}a^{-4}-9z^{4}+5a^{3}z^{3}+4az^{3}+2z^{3}a^{-1}-3z^{3}a^{-3}-6z^{3}a^{-5}+3a^{2}z^{2}+13z^{2}a^{-2}+9z^{2}a^{-4}+7z^{2}-5a^{3}z-7az-3za^{-1}+za^{-5}-3a^{2}-a^{-2}-3+2a^{3}z^{-1}+3az^{-1}+a^{-1}z^{-1}$ (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

6

χ

12

1

10

1

-1

8

1

0

6

2

1

-1

4

1

-1

2

1

2

1

0

1

3

1

-2

2

-4

1

0

-6

1

-8

1

Integral Khovanov Homology

(db, data source)

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