L11n443

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_18,11,19,12 X_3,11,4,10 X_9,1,10,4 X_7,17,8,16 X_15,5,16,8 X_20,14,21,13 X_22,19,15,20 X_12,22,13,21 X_14,17,9,18
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, 3, -10, 8, -11}, {-7, 6, 11, -3, 9, -8, 10, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-uvw-uvx+uv-uw^{2}x+uw^{2}+2uwx-uw-vwx^{2}+2vwx+vx^{2}-vx+w^{2}x^{2}-w^{2}x-wx^{2}}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	$q^{9/2}+{\frac {1}{q^{9/2}}}-2q^{7/2}-{\frac {2}{q^{7/2}}}-q^{5/2}+{\frac {2}{q^{5/2}}}-q^{3/2}-{\frac {3}{q^{3/2}}}-q^{11/2}-3{\sqrt {q}}+{\frac {1}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$az^{5}-z^{5}a^{-1}-a^{3}z^{3}+5az^{3}-7z^{3}a^{-1}+2z^{3}a^{-3}-2a^{3}z+9az-14za^{-1}+8za^{-3}-za^{-5}-a^{3}z^{-1}+6az^{-1}-11a^{-1}z^{-1}+8a^{-3}z^{-1}-2a^{-5}z^{-1}+az^{-3}-3a^{-1}z^{-3}+3a^{-3}z^{-3}-a^{-5}z^{-3}$ (db)
Kauffman polynomial	$z^{7}a^{-5}-6z^{5}a^{-5}+11z^{3}a^{-5}-a^{-5}z^{-3}-10za^{-5}+5a^{-5}z^{-1}+z^{8}a^{-4}+a^{4}z^{6}-5z^{6}a^{-4}-4a^{4}z^{4}+3z^{4}a^{-4}+3a^{4}z^{2}+7z^{2}a^{-4}+3a^{-4}z^{-2}-a^{4}-9a^{-4}+2a^{3}z^{7}+3z^{7}a^{-3}-9a^{3}z^{5}-21z^{5}a^{-3}+10a^{3}z^{3}+42z^{3}a^{-3}-3a^{-3}z^{-3}-6a^{3}z-35za^{-3}+2a^{3}z^{-1}+14a^{-3}z^{-1}+a^{2}z^{8}+z^{8}a^{-2}-2a^{2}z^{6}-5z^{6}a^{-2}-8a^{2}z^{4}-3z^{4}a^{-2}+14a^{2}z^{2}+24z^{2}a^{-2}+6a^{-2}z^{-2}-6a^{2}-21a^{-2}+4az^{7}+4z^{7}a^{-1}-23az^{5}-29z^{5}a^{-1}+39az^{3}+60z^{3}a^{-1}-az^{-3}-3a^{-1}z^{-3}-30az-49za^{-1}+11az^{-1}+18a^{-1}z^{-1}+z^{8}-3z^{6}-10z^{4}+28z^{2}+3z^{-2}-18$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

12

1

10

0

8

2

1

6

3

1

3

4

1

4

1

2

4

2

4

0

2

5

1

2

-2

2

1

2

-4

1

3

1

-6

1

0

-8

1

-10

1

-1

Integral Khovanov Homology

(db, data source)

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