L10a21

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_18,15,19,16 X_16,10,17,9 X_8,18,9,17 X_4,19,1,20 X_12,6,13,5 X_10,4,11,3 X_20,12,5,11 X_2,14,3,13
Gauss code	{1, -10, 8, -6}, {7, -1, 2, -5, 4, -8, 9, -7, 10, -2, 3, -4, 5, -3, 6, -9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)^{3}\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-8q^{9/2}+12q^{7/2}-{\frac {1}{q^{7/2}}}-15q^{5/2}+{\frac {4}{q^{5/2}}}+16q^{3/2}-{\frac {8}{q^{3/2}}}-q^{13/2}+4q^{11/2}-16{\sqrt {q}}+{\frac {11}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+az^{5}-4z^{5}a^{-1}+2z^{5}a^{-3}+2az^{3}-6z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}+az-3za^{-1}+3za^{-3}-za^{-5}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{9}a^{-1}-2z^{9}a^{-3}-12z^{8}a^{-2}-6z^{8}a^{-4}-6z^{8}-7az^{7}-12z^{7}a^{-1}-12z^{7}a^{-3}-7z^{7}a^{-5}-4a^{2}z^{6}+19z^{6}a^{-2}+5z^{6}a^{-4}-4z^{6}a^{-6}+6z^{6}-a^{3}z^{5}+13az^{5}+32z^{5}a^{-1}+31z^{5}a^{-3}+12z^{5}a^{-5}-z^{5}a^{-7}+6a^{2}z^{4}-7z^{4}a^{-2}+4z^{4}a^{-4}+6z^{4}a^{-6}+z^{4}+a^{3}z^{3}-8az^{3}-25z^{3}a^{-1}-23z^{3}a^{-3}-6z^{3}a^{-5}+z^{3}a^{-7}-a^{2}z^{2}-z^{2}a^{-2}-3z^{2}a^{-4}-2z^{2}a^{-6}-z^{2}+2az+6za^{-1}+6za^{-3}+2za^{-5}+1-az^{-1}-a^{-1}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

3

-3

10

5

1

4

8

7

3

-4

6

8

5

3

4

8

7

-1

2

8

0

5

10

5

-2

3

6

-3

-4

1

5

4

-6

3

-3

-8

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-4$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$