L10a29

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_14,8,15,7 X_20,12,5,11 X_8,20,9,19 X_16,9,17,10 X_18,15,19,16 X_10,17,11,18 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -5, 6, -8, 4, -2, 10, -3, 7, -6, 8, -7, 5, -4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(v^{2}-v+1\right)^{2}}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{11/2}-3q^{9/2}+6q^{7/2}-9q^{5/2}+11q^{3/2}-13{\sqrt {q}}+{\frac {10}{\sqrt {q}}}-{\frac {9}{q^{3/2}}}+{\frac {6}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-3}-a^{3}z^{3}+3z^{3}a^{-3}-2a^{3}z+3za^{-3}-a^{3}z^{-1}+a^{-3}z^{-1}-z^{7}a^{-1}+2az^{5}-5z^{5}a^{-1}+7az^{3}-10z^{3}a^{-1}+8az-9za^{-1}+4az^{-1}-4a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}-z^{2}a^{-6}+3z^{5}a^{-5}-3z^{3}a^{-5}+a^{4}z^{6}+5z^{6}a^{-4}-3a^{4}z^{4}-6z^{4}a^{-4}+3a^{4}z^{2}+3z^{2}a^{-4}-a^{4}-a^{-4}+3a^{3}z^{7}+6z^{7}a^{-3}-9a^{3}z^{5}-10z^{5}a^{-3}+8a^{3}z^{3}+11z^{3}a^{-3}-4a^{3}z-6za^{-3}+a^{3}z^{-1}+a^{-3}z^{-1}+3a^{2}z^{8}+4z^{8}a^{-2}-4a^{2}z^{6}-z^{6}a^{-2}-8a^{2}z^{4}-9z^{4}a^{-2}+11a^{2}z^{2}+12z^{2}a^{-2}-4a^{2}-4a^{-2}+az^{9}+z^{9}a^{-1}+8az^{7}+11z^{7}a^{-1}-31az^{5}-35z^{5}a^{-1}+32az^{3}+38z^{3}a^{-1}-15az-17za^{-1}+4az^{-1}+4a^{-1}z^{-1}+7z^{8}-11z^{6}-7z^{4}+16z^{2}-7$ (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

5

2

3

4

6

4

-2

2

7

5

2

0

5

8

3

-2

4

5

-1

-4

2

5

3

-6

1

4

-3

-8

2

-10

1

-1

Integral Khovanov Homology

(db, data source)

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