L11n23

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_17,1,18,4 X_9,21,10,20 X₃₈₄₉ X_21,18,22,19 X_11,14,12,15 X_5,13,6,12 X_13,5,14,22 X_19,11,20,10 X_2,16,3,15
Gauss code	{1, -11, -5, 3}, {-8, -1, 2, 5, -4, 10, -7, 8, -9, 7, 11, -2, -3, 6, -10, 4, -6, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(v^{2}-4v+1\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-q^{9/2}+{\frac {1}{q^{9/2}}}+3q^{7/2}-{\frac {3}{q^{7/2}}}-6q^{5/2}+{\frac {4}{q^{5/2}}}+7q^{3/2}-{\frac {7}{q^{3/2}}}-8{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-a^{-5}z^{-1}-a^{3}z^{3}-a^{3}z+4za^{-3}+3a^{-3}z^{-1}+az^{5}+3az^{3}-4z^{3}a^{-1}+5az-8za^{-1}+2az^{-1}-4a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-3a^{2}z^{8}-3z^{8}a^{-2}-6z^{8}-3a^{3}z^{7}-4az^{7}-3z^{7}a^{-1}-2z^{7}a^{-3}-a^{4}z^{6}+9a^{2}z^{6}+12z^{6}a^{-2}+22z^{6}+11a^{3}z^{5}+26az^{5}+22z^{5}a^{-1}+7z^{5}a^{-3}+3a^{4}z^{4}-5a^{2}z^{4}-23z^{4}a^{-2}-3z^{4}a^{-4}-28z^{4}-10a^{3}z^{3}-34az^{3}-40z^{3}a^{-1}-17z^{3}a^{-3}-z^{3}a^{-5}-a^{4}z^{2}+2a^{2}z^{2}+15z^{2}a^{-2}+3z^{2}a^{-4}+15z^{2}+3a^{3}z+15az+24za^{-1}+14za^{-3}+2za^{-5}-a^{2}-3a^{-2}-a^{-4}-2-2az^{-1}-4a^{-1}z^{-1}-3a^{-3}z^{-1}-a^{-5}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

χ

10

1

8

2

-2

6

4

1

3

4

3

2

-1

2

5

4

1

0

5

0

-2

2

3

-1

-4

2

5

3

-6

1

2

-1

-8

2

-10

1

-1

Integral Khovanov Homology

(db, data source)

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