L10a71

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_18,9,19,10 X_14,6,15,5 X_16,8,17,7 X_10,4,11,3 X_20,12,7,11 X_12,15,13,16 X_6,14,1,13 X_4,19,5,20 X_2,18,3,17
Gauss code	{1, -10, 5, -9, 3, -8}, {4, -1, 2, -5, 6, -7, 8, -3, 7, -4, 10, -2, 9, -6}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{4}-4u^{2}v^{3}+5u^{2}v^{2}-3u^{2}v+u^{2}-2uv^{4}+7uv^{3}-11uv^{2}+7uv-2u+v^{4}-3v^{3}+5v^{2}-4v+1}{uv^{2}}}$ (db)
Jones polynomial	$-q^{13/2}+4q^{11/2}-9q^{9/2}+14q^{7/2}-18q^{5/2}+19q^{3/2}-19{\sqrt {q}}+{\frac {14}{\sqrt {q}}}-{\frac {10}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+az^{5}-3z^{5}a^{-1}+2z^{5}a^{-3}+az^{3}-2z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}-az+2za^{-1}+za^{-3}-za^{-5}+a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-3z^{9}a^{-1}-3z^{9}a^{-3}-17z^{8}a^{-2}-8z^{8}a^{-4}-9z^{8}-10az^{7}-17z^{7}a^{-1}-15z^{7}a^{-3}-8z^{7}a^{-5}-5a^{2}z^{6}+25z^{6}a^{-2}+8z^{6}a^{-4}-4z^{6}a^{-6}+8z^{6}-a^{3}z^{5}+16az^{5}+42z^{5}a^{-1}+39z^{5}a^{-3}+13z^{5}a^{-5}-z^{5}a^{-7}+5a^{2}z^{4}-4z^{4}a^{-2}+z^{4}a^{-4}+5z^{4}a^{-6}+5z^{4}-6az^{3}-22z^{3}a^{-1}-26z^{3}a^{-3}-9z^{3}a^{-5}+z^{3}a^{-7}-2z^{2}a^{-2}-2z^{2}a^{-4}-2z^{2}a^{-6}-2z^{2}-az+4za^{-3}+3za^{-5}-a^{-2}+a^{-1}z^{-1}+a^{-3}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

3

-3

10

6

1

5

8

3

-5

6

10

6

4

10

9

-1

2

9

0

6

11

5

-2

4

8

-4

1

6

5

-6

4

-4

-8

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-4$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$