L10a51

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_20,10,7,9 X₂₇₃₈ X_4,15,5,16 X_12,5,13,6 X_16,12,17,11 X_6,18,1,17 X_14,19,15,20 X_18,13,19,14
Gauss code	{1, -4, 2, -5, 6, -8}, {4, -1, 3, -2, 7, -6, 10, -9, 5, -7, 8, -10, 9, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {\left(t(1)t(2)^{2}-t(2)^{2}-2t(1)t(2)+t(2)+t(1)-1\right)\left(t(1)t(2)^{2}-t(2)^{2}-t(1)t(2)+2t(2)+t(1)-1\right)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}-q^{7/2}+{\frac {12}{q^{7/2}}}+4q^{5/2}-{\frac {16}{q^{5/2}}}-8q^{3/2}+{\frac {16}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+12{\sqrt {q}}-{\frac {16}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z-2a^{3}z^{5}-5a^{3}z^{3}-3a^{3}z+a^{3}z^{-1}+az^{7}+4az^{5}-z^{5}a^{-1}+6az^{3}-2z^{3}a^{-1}+2az-za^{-1}-az^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-a^{7}z^{3}+4a^{6}z^{6}-6a^{6}z^{4}+2a^{6}z^{2}+7a^{5}z^{7}-12a^{5}z^{5}+7a^{5}z^{3}-2a^{5}z+6a^{4}z^{8}-4a^{4}z^{6}-6a^{4}z^{4}+4a^{4}z^{2}+2a^{3}z^{9}+13a^{3}z^{7}-33a^{3}z^{5}+z^{5}a^{-3}+24a^{3}z^{3}-z^{3}a^{-3}-5a^{3}z-a^{3}z^{-1}+12a^{2}z^{8}-16a^{2}z^{6}+4z^{6}a^{-2}-6z^{4}a^{-2}+4a^{2}z^{2}+2z^{2}a^{-2}+a^{2}+2az^{9}+13az^{7}+7z^{7}a^{-1}-33az^{5}-12z^{5}a^{-1}+24az^{3}+7z^{3}a^{-1}-5az-2za^{-1}-az^{-1}+6z^{8}-4z^{6}-6z^{4}+4z^{2}$ (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

-3

4

5

1

4

2

7

3

-4

0

9

5

4

-2

8

0

-4

8

0

-6

5

9

4

-8

3

7

-4

-10

1

5

4

-12

3

-3

-14

1

Integral Khovanov Homology

(db, data source)

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