L11n121

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Link Presentations

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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_17,5,18,22 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 $u$ , $v$ , $w$ , ...)	$-{\frac {-uv^{4}+2uv^{3}-2uv^{2}+uv+u+v^{5}+v^{4}-2v^{3}+2v^{2}-v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{13/2}+3q^{11/2}-3q^{9/2}+4q^{7/2}-4q^{5/2}+3q^{3/2}-3{\sqrt {q}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}-za^{-5}+z^{5}a^{-3}+4z^{3}a^{-3}+6za^{-3}+2a^{-3}z^{-1}-z^{5}a^{-1}+az^{3}-7z^{3}a^{-1}+4az-11za^{-1}+3az^{-1}-5a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-az^{7}-z^{7}a^{-3}-2z^{7}a^{-5}-z^{6}a^{-2}-4z^{6}a^{-4}-3z^{6}a^{-6}+7az^{5}+3z^{5}a^{-1}+2z^{5}a^{-3}+5z^{5}a^{-5}-z^{5}a^{-7}+4z^{4}a^{-2}+10z^{4}a^{-4}+9z^{4}a^{-6}+3z^{4}-14az^{3}-14z^{3}a^{-1}-4z^{3}a^{-3}-2z^{3}a^{-5}+2z^{3}a^{-7}-10z^{2}a^{-2}-5z^{2}a^{-4}-4z^{2}a^{-6}-9z^{2}+10az+14za^{-1}+5za^{-3}+za^{-5}+5a^{-2}-a^{-6}+5-3az^{-1}-5a^{-1}z^{-1}-2a^{-3}z^{-1}$ (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

14

1

12

2

-2

10

1

0

8

3

2

-1

6

1

2

1

0

4

2

3

1

2

1

3

2

0

3

-2

1

0

-4

1

-6

1

Integral Khovanov Homology

(db, data source)

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