L11n245

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X_10,11,1,12 X_14,5,15,6 X_9,19,10,18 X_17,3,18,2 X_16,8,17,7 X₃₈₄₉ X_20,16,21,15 X_22,13,11,14 X_4,20,5,19 X_6,21,7,22
Gauss code	{1, 5, -7, -10, 3, -11, 6, 7, -4, -2}, {2, -1, 9, -3, 8, -6, -5, 4, 10, -8, 11, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(2u^{2}v-u^{2}+uv^{2}-uv+u-v^{2}+2v\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {2}{q^{9/2}}}-3q^{7/2}+{\frac {6}{q^{7/2}}}+6q^{5/2}-{\frac {9}{q^{5/2}}}-10q^{3/2}+{\frac {11}{q^{3/2}}}+11{\sqrt {q}}-{\frac {13}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-2az^{5}-z^{5}a^{-1}+2a^{3}z^{3}-5az^{3}-z^{3}a^{-1}+z^{3}a^{-3}+2a^{3}z-3az+za^{-3}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2az^{9}-2z^{9}a^{-1}-5a^{2}z^{8}-4z^{8}a^{-2}-9z^{8}-4a^{3}z^{7}-3az^{7}-2z^{7}a^{-1}-3z^{7}a^{-3}-a^{4}z^{6}+10a^{2}z^{6}+10z^{6}a^{-2}-z^{6}a^{-4}+22z^{6}+3a^{3}z^{5}+10az^{5}+16z^{5}a^{-1}+9z^{5}a^{-3}-6a^{4}z^{4}-14a^{2}z^{4}-5z^{4}a^{-2}+3z^{4}a^{-4}-16z^{4}-3a^{5}z^{3}-2a^{3}z^{3}-4az^{3}-13z^{3}a^{-1}-8z^{3}a^{-3}+5a^{4}z^{2}+8a^{2}z^{2}-2z^{2}a^{-4}+5z^{2}+a^{5}z+4za^{-1}+3za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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

χ

10

1

-1

8

2

6

4

1

-3

4

6

2

4

2

5

4

-1

0

8

6

2

-2

5

7

2

-4

4

6

-2

-6

2

5

3

-8

4

-4

-10

2

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-4$	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=0$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$