L10a4

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_12,6,13,5 X₈₄₉₃ X_16,10,17,9 X_20,18,5,17 X_18,11,19,12 X_10,19,11,20 X_2,14,3,13
Gauss code	{1, -10, 5, -3}, {4, -1, 2, -5, 6, -9, 8, -4, 10, -2, 3, -6, 7, -8, 9, -7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(v^{4}-3v^{3}+3v^{2}-3v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+7q^{7/2}-11q^{5/2}+14q^{3/2}-15{\sqrt {q}}+{\frac {13}{\sqrt {q}}}-{\frac {11}{q^{3/2}}}+{\frac {7}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-a^{3}z^{3}+5az^{3}-5z^{3}a^{-1}+2z^{3}a^{-3}-a^{3}z+az+a^{3}z^{-1}-2az^{-1}+2a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-2az^{9}-2z^{9}a^{-1}-5a^{2}z^{8}-6z^{8}a^{-2}-11z^{8}-4a^{3}z^{7}-7az^{7}-11z^{7}a^{-1}-8z^{7}a^{-3}-a^{4}z^{6}+11a^{2}z^{6}+2z^{6}a^{-2}-7z^{6}a^{-4}+21z^{6}+11a^{3}z^{5}+29az^{5}+30z^{5}a^{-1}+8z^{5}a^{-3}-4z^{5}a^{-5}+2a^{4}z^{4}-3a^{2}z^{4}+9z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}-4z^{4}-8a^{3}z^{3}-21az^{3}-17z^{3}a^{-1}-z^{3}a^{-3}+3z^{3}a^{-5}-a^{4}z^{2}-2a^{2}z^{2}-6z^{2}a^{-2}-2z^{2}a^{-4}-5z^{2}+a^{3}z-2za^{-1}-za^{-3}+1+a^{3}z^{-1}+2az^{-1}+2a^{-1}z^{-1}+a^{-3}z^{-1}$ (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

3

8

4

1

-3

6

7

3

4

7

4

-3

2

8

7

1

0

7

9

2

-2

4

6

-2

-4

3

7

4

-6

1

4

-3

-8

3

-10

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$