L10a168

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(x-1)\left(uvwx-uvw-2uvx-2uwx+uw-ux^{2}+2ux-2vwx+vw-vx^{2}+2vx+2wx+x^{2}-x\right)}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}x^{3/2}}}$ (db)
Jones polynomial	$-{\frac {6}{q^{9/2}}}-q^{7/2}+{\frac {9}{q^{7/2}}}+3q^{5/2}-{\frac {14}{q^{5/2}}}-8q^{3/2}+{\frac {12}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}+10{\sqrt {q}}-{\frac {14}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{-1}+a^{5}z^{-3}-3a^{5}z-a^{5}z^{-1}+3a^{3}z^{3}-3a^{3}z^{-3}-4a^{3}z^{-1}-za^{-3}-az^{5}+az^{3}+3az^{-3}+2z^{3}a^{-1}-a^{-1}z^{-3}+5az+7az^{-1}-za^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-3a^{4}z^{8}-7a^{2}z^{8}-4z^{8}-3a^{5}z^{7}-10a^{3}z^{7}-13az^{7}-6z^{7}a^{-1}-2a^{6}z^{6}+3a^{2}z^{6}-3z^{6}a^{-2}-2z^{6}-a^{7}z^{5}+3a^{5}z^{5}+27a^{3}z^{5}+37az^{5}+13z^{5}a^{-1}-z^{5}a^{-3}+3a^{6}z^{4}+8a^{4}z^{4}+21a^{2}z^{4}+4z^{4}a^{-2}+20z^{4}+3a^{7}z^{3}-a^{5}z^{3}-32a^{3}z^{3}-45az^{3}-15z^{3}a^{-1}+2z^{3}a^{-3}-15a^{4}z^{2}-36a^{2}z^{2}-z^{2}a^{-2}-22z^{2}-3a^{7}z+3a^{5}z+24a^{3}z+33az+14za^{-1}-za^{-3}-a^{6}+11a^{4}+24a^{2}+13+a^{7}z^{-1}-3a^{5}z^{-1}-12a^{3}z^{-1}-14az^{-1}-6a^{-1}z^{-1}-3a^{4}z^{-2}-6a^{2}z^{-2}-3z^{-2}+a^{5}z^{-3}+3a^{3}z^{-3}+3az^{-3}+a^{-1}z^{-3}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

3

1

-2

4

5

2

5

3

-2

0

9

5

4

-2

7

9

2

-4

7

5

2

-6

2

7

5

-8

4

7

-3

-10

1

5

4

-12

1

-1

-14

1

Integral Khovanov Homology

(db, data source)

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