L10a103

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_12,4,13,3 X_20,12,9,11 X_2,9,3,10 X_4,17,5,18 X_18,5,19,6 X_6,14,7,13 X_14,8,15,7 X_8,16,1,15 X_16,19,17,20
Gauss code	{1, -4, 2, -5, 6, -7, 8, -9}, {4, -1, 3, -2, 7, -8, 9, -10, 5, -6, 10, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(2)t(1)-t(1)+1)(t(1)t(2)-t(2)+1)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-3q^{9/2}+{\frac {1}{q^{9/2}}}+6q^{7/2}-{\frac {3}{q^{7/2}}}-10q^{5/2}+{\frac {5}{q^{5/2}}}+11q^{3/2}-{\frac {9}{q^{3/2}}}+q^{11/2}-12{\sqrt {q}}+{\frac {11}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-5z^{5}a^{-1}+z^{5}a^{-3}-a^{3}z^{3}+7az^{3}-10z^{3}a^{-1}+3z^{3}a^{-3}-2a^{3}z+7az-8za^{-1}+3za^{-3}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-3a^{2}z^{8}-4z^{8}a^{-2}-7z^{8}-3a^{3}z^{7}-7az^{7}-10z^{7}a^{-1}-6z^{7}a^{-3}-a^{4}z^{6}+6a^{2}z^{6}+z^{6}a^{-2}-5z^{6}a^{-4}+13z^{6}+10a^{3}z^{5}+27az^{5}+28z^{5}a^{-1}+8z^{5}a^{-3}-3z^{5}a^{-5}+3a^{4}z^{4}+5z^{4}a^{-2}+5z^{4}a^{-4}-z^{4}a^{-6}-4z^{4}-10a^{3}z^{3}-27az^{3}-26z^{3}a^{-1}-6z^{3}a^{-3}+3z^{3}a^{-5}-2a^{4}z^{2}-2a^{2}z^{2}-3z^{2}a^{-2}-2z^{2}a^{-4}+z^{2}a^{-6}+3a^{3}z+10az+10za^{-1}+2za^{-3}-za^{-5}+1-az^{-1}-a^{-1}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

2

8

4

1

-3

6

2

4

5

4

-1

2

7

6

1

0

6

7

1

-2

3

5

-2

-4

2

6

4

-6

1

3

-2

-8

2

-10

1

-1

Integral Khovanov Homology

(db, data source)

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