L10a104

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Link Presentations

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{3}\left(-v^{3}\right)+2u^{3}v^{2}-u^{3}v+3u^{2}v^{3}-7u^{2}v^{2}+7u^{2}v-2u^{2}-2uv^{3}+7uv^{2}-7uv+3u-v^{2}+2v-1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-7q^{9/2}+11q^{7/2}-{\frac {1}{q^{7/2}}}-15q^{5/2}+{\frac {4}{q^{5/2}}}+15q^{3/2}-{\frac {8}{q^{3/2}}}-q^{13/2}+3q^{11/2}-15{\sqrt {q}}+{\frac {12}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}-2za^{-5}-a^{-5}z^{-1}+2z^{5}a^{-3}+6z^{3}a^{-3}+6za^{-3}+a^{-3}z^{-1}-z^{7}a^{-1}+az^{5}-4z^{5}a^{-1}+2az^{3}-6z^{3}a^{-1}+az-3za^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-2z^{3}a^{-7}+za^{-7}+3z^{6}a^{-6}-5z^{4}a^{-6}+2z^{2}a^{-6}+5z^{7}a^{-5}-8z^{5}a^{-5}+6z^{3}a^{-5}-4za^{-5}+a^{-5}z^{-1}+5z^{8}a^{-4}-5z^{6}a^{-4}+2z^{2}a^{-4}-a^{-4}+2z^{9}a^{-3}+9z^{7}a^{-3}+a^{3}z^{5}-25z^{5}a^{-3}-a^{3}z^{3}+23z^{3}a^{-3}-9za^{-3}+a^{-3}z^{-1}+11z^{8}a^{-2}+4a^{2}z^{6}-17z^{6}a^{-2}-6a^{2}z^{4}+6z^{4}a^{-2}+2a^{2}z^{2}+z^{2}a^{-2}+2z^{9}a^{-1}+7az^{7}+11z^{7}a^{-1}-12az^{5}-29z^{5}a^{-1}+6az^{3}+22z^{3}a^{-1}-2az-6za^{-1}+6z^{8}-5z^{6}-5z^{4}+3z^{2}$ (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

2

-2

10

5

1

4

8

7

3

-4

6

8

4

7

0

2

8

0

5

8

3

-2

3

7

-4

1

5

4

-6

3

-3

-8

1

Integral Khovanov Homology

(db, data source)

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