L10n113

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_11,19,12,18 X_3,11,4,10 X_9,1,10,4 X_7,15,8,14 X_13,5,14,8 X_20,15,17,16 X_16,19,13,20 X_17,9,18,12
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 10}, {-7, 6, 8, -9}, {-10, 3, 9, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-uvw+uvxy-uvy+uv+uwy-uxy+vw-vx-wxy+wx-w+xy}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}{\sqrt {x}}{\sqrt {y}}}}$ (db)
Jones polynomial	$q^{6}-q^{5}+5q^{4}-q^{3}+q^{-3}+5q^{2}+q^{-1}+5$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{-6}z^{-4}+2a^{-6}z^{-2}+a^{-6}-4a^{-4}z^{-4}-3z^{2}a^{-4}-9a^{-4}z^{-2}-8a^{-4}+2z^{4}a^{-2}+a^{2}z^{-4}+6a^{-2}z^{-4}+a^{2}z^{2}+9z^{2}a^{-2}+3a^{2}z^{-2}+15a^{-2}z^{-2}+3a^{2}+16a^{-2}-z^{4}-4z^{-4}-7z^{2}-11z^{-2}-12$ (db)
Kauffman polynomial	$z^{6}a^{-6}-5z^{4}a^{-6}-a^{-6}z^{-4}+10z^{2}a^{-6}+5a^{-6}z^{-2}-10a^{-6}+z^{7}a^{-5}-z^{5}a^{-5}-10z^{3}a^{-5}+4a^{-5}z^{-3}+20za^{-5}-15a^{-5}z^{-1}+6z^{6}a^{-4}-25z^{4}a^{-4}-4a^{-4}z^{-4}+30z^{2}a^{-4}+14a^{-4}z^{-2}-25a^{-4}+z^{7}a^{-3}+3z^{5}a^{-3}-30z^{3}a^{-3}+12a^{-3}z^{-3}+55za^{-3}-41a^{-3}z^{-1}+a^{2}z^{6}+5z^{6}a^{-2}-6a^{2}z^{4}-25z^{4}a^{-2}-a^{2}z^{-4}-6a^{-2}z^{-4}+10a^{2}z^{2}+40z^{2}a^{-2}+5a^{2}z^{-2}+18a^{-2}z^{-2}-10a^{2}-31a^{-2}+az^{5}+5z^{5}a^{-1}-10az^{3}-30z^{3}a^{-1}+4az^{-3}+12a^{-1}z^{-3}+20az+55za^{-1}-15az^{-1}-41a^{-1}z^{-1}+z^{6}-11z^{4}-4z^{-4}+30z^{2}+14z^{-2}-25$ (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

13

1

11

0

9

5

1

4

7

1

5

4

5

4

3

4

1

5

1

10

4

5

-1

6

-3

1

-5

1

-7

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$	$i=3$
$r=-4$		${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-3$
$r=-2$		${\mathbb {Z} }$
$r=-1$		${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }^{6}$	${\mathbb {Z} }^{10}$	${\mathbb {Z} }^{4}$
$r=1$	${\mathbb {Z} }^{4}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=4$	${\mathbb {Z} }^{5}$	${\mathbb {Z} }^{5}$
$r=5$	${\mathbb {Z} }$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$