L10a109

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X_16,8,17,7 X_10,5,1,6 X₆₃₇₄ X_4,9,5,10 X_18,14,19,13 X_20,16,11,15 X_14,20,15,19 X_2,11,3,12 X_8,18,9,17
Gauss code	{1, -9, 4, -5, 3, -4, 2, -10, 5, -3}, {9, -1, 6, -8, 7, -2, 10, -6, 8, -7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(u^{2}v^{2}+u^{2}-2uv^{2}+uv-2u+v^{2}+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{11/2}-3q^{9/2}+7q^{7/2}-10q^{5/2}+11q^{3/2}-13{\sqrt {q}}+{\frac {10}{\sqrt {q}}}-{\frac {9}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {2}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-3}-a^{3}z^{3}+3z^{3}a^{-3}-3a^{3}z+4za^{-3}-2a^{3}z^{-1}+2a^{-3}z^{-1}-z^{7}a^{-1}+2az^{5}-5z^{5}a^{-1}+8az^{3}-11z^{3}a^{-1}+12az-13za^{-1}+7az^{-1}-7a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}-z^{2}a^{-6}+3z^{5}a^{-5}-2z^{3}a^{-5}+a^{4}z^{6}+6z^{6}a^{-4}-4a^{4}z^{4}-8z^{4}a^{-4}+5a^{4}z^{2}+6z^{2}a^{-4}-2a^{4}-2a^{-4}+2a^{3}z^{7}+7z^{7}a^{-3}-6a^{3}z^{5}-11z^{5}a^{-3}+6a^{3}z^{3}+10z^{3}a^{-3}-4a^{3}z-6za^{-3}+2a^{3}z^{-1}+2a^{-3}z^{-1}+2a^{2}z^{8}+4z^{8}a^{-2}-a^{2}z^{6}+2z^{6}a^{-2}-11a^{2}z^{4}-17z^{4}a^{-2}+17a^{2}z^{2}+20z^{2}a^{-2}-8a^{2}-8a^{-2}+az^{9}+z^{9}a^{-1}+5az^{7}+10z^{7}a^{-1}-19az^{5}-27z^{5}a^{-1}+21az^{3}+27z^{3}a^{-1}-16az-18za^{-1}+7az^{-1}+7a^{-1}z^{-1}+6z^{8}-6z^{6}-15z^{4}+25z^{2}-13$ (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

5

1

-4

6

5

2

3

4

6

5

-1

2

7

5

2

0

5

8

3

-2

4

5

-1

-4

1

5

4

-6

1

4

-3

-8

1

-10

1

-1

Integral Khovanov Homology

(db, data source)

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