L10a28

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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_20,16,5,15 X_16,9,17,10 X_8,19,9,20 X_18,11,19,12 X_10,17,11,18 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -6, 5, -8, 7, -2, 10, -3, 4, -5, 8, -7, 6, -4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}-t(2)+1\right)^{2}}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+3q^{5/2}-6q^{3/2}+8{\sqrt {q}}-{\frac {12}{\sqrt {q}}}+{\frac {12}{q^{3/2}}}-{\frac {11}{q^{5/2}}}+{\frac {9}{q^{7/2}}}-{\frac {6}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+2a^{5}z+a^{5}z^{-1}-2a^{3}z^{5}-7a^{3}z^{3}-8a^{3}z-3a^{3}z^{-1}+az^{7}+5az^{5}-z^{5}a^{-1}+10az^{3}-3z^{3}a^{-1}+9az-3za^{-1}+4az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-3a^{4}z^{8}-7a^{2}z^{8}-4z^{8}-4a^{5}z^{7}-8a^{3}z^{7}-9az^{7}-5z^{7}a^{-1}-3a^{6}z^{6}+13a^{2}z^{6}-3z^{6}a^{-2}+7z^{6}-a^{7}z^{5}+6a^{5}z^{5}+24a^{3}z^{5}+31az^{5}+13z^{5}a^{-1}-z^{5}a^{-3}+6a^{6}z^{4}+9a^{4}z^{4}-7a^{2}z^{4}+6z^{4}a^{-2}-4z^{4}+2a^{7}z^{3}-a^{5}z^{3}-28a^{3}z^{3}-40az^{3}-13z^{3}a^{-1}+2z^{3}a^{-3}-3a^{6}z^{2}-8a^{4}z^{2}-5a^{2}z^{2}-z^{2}a^{-2}-z^{2}-a^{7}z+2a^{5}z+15a^{3}z+19az+7za^{-1}+a^{6}+3a^{4}+3a^{2}+2-a^{5}z^{-1}-3a^{3}z^{-1}-4az^{-1}-2a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

2

-2

4

1

3

2

4

2

-2

0

8

4

-2

6

0

-4

5

6

-1

-6

4

6

2

-8

2

5

-3

-10

1

4

3

-12

2

-2

-14

1

Integral Khovanov Homology

(db, data source)

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