L10a50

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Link Presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {6uv^{2}-7uv+2u+2v^{3}-7v^{2}+6v}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {7}{q^{9/2}}}-{\frac {10}{q^{11/2}}}+{\frac {9}{q^{13/2}}}-{\frac {9}{q^{15/2}}}+{\frac {7}{q^{17/2}}}-{\frac {4}{q^{19/2}}}+{\frac {3}{q^{21/2}}}-{\frac {1}{q^{23/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{11}z-a^{11}z^{-1}-a^{9}z^{3}+2a^{9}z+2a^{9}z^{-1}-3a^{7}z^{3}-2a^{7}z-3a^{5}z^{3}-3a^{5}z-a^{5}z^{-1}-a^{3}z^{3}$ (db)
Kauffman polynomial	$-z^{7}a^{13}+4z^{5}a^{13}-4z^{3}a^{13}+za^{13}-3z^{8}a^{12}+14z^{6}a^{12}-19z^{4}a^{12}+6z^{2}a^{12}+2a^{12}-2z^{9}a^{11}+5z^{7}a^{11}+3z^{5}a^{11}-9z^{3}a^{11}+2za^{11}-a^{11}z^{-1}-8z^{8}a^{10}+30z^{6}a^{10}-29z^{4}a^{10}+2z^{2}a^{10}+5a^{10}-2z^{9}a^{9}-z^{7}a^{9}+17z^{5}a^{9}-15z^{3}a^{9}+4za^{9}-2a^{9}z^{-1}-5z^{8}a^{8}+9z^{6}a^{8}+z^{4}a^{8}-4z^{2}a^{8}+3a^{8}-7z^{7}a^{7}+12z^{5}a^{7}-3z^{3}a^{7}-7z^{6}a^{6}+8z^{4}a^{6}-a^{6}-6z^{5}a^{5}+6z^{3}a^{5}-3za^{5}+a^{5}z^{-1}-3z^{4}a^{4}-z^{3}a^{3}$ (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		\

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

1

-4

3

1

-2

-6

3

-8

4

3

-1

-10

6

3

-12

4

5

1

-14

5

0

-16

2

4

2

-18

2

5

-3

-20

1

2

1

-22

2

-2

-24

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-10$	${\mathbb {Z} }$
$r=-9$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-7$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=-3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$