L10a52

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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_16,11,17,12 X_12,5,13,6 X_4,18,5,17 X_14,20,15,19 X_18,14,19,13 X_6,15,1,16
Gauss code	{1, -4, 2, -7, 6, -10}, {4, -1, 3, -2, 5, -6, 9, -8, 10, -5, 7, -9, 8, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}+t(2)^{4}-3t(1)^{2}t(2)^{3}+6t(1)t(2)^{3}-3t(2)^{3}+4t(1)^{2}t(2)^{2}-7t(1)t(2)^{2}+4t(2)^{2}-3t(1)^{2}t(2)+6t(1)t(2)-3t(2)+t(1)^{2}-2t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-4q^{9/2}+{\frac {1}{q^{9/2}}}+8q^{7/2}-{\frac {4}{q^{7/2}}}-12q^{5/2}+{\frac {7}{q^{5/2}}}+15q^{3/2}-{\frac {12}{q^{3/2}}}+q^{11/2}-16{\sqrt {q}}+{\frac {14}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-a^{3}z^{3}+5az^{3}-6z^{3}a^{-1}+2z^{3}a^{-3}-a^{3}z+2az-3za^{-1}+za^{-3}+a^{3}z^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-2az^{9}-2z^{9}a^{-1}-5a^{2}z^{8}-7z^{8}a^{-2}-12z^{8}-4a^{3}z^{7}-9az^{7}-15z^{7}a^{-1}-10z^{7}a^{-3}-a^{4}z^{6}+10a^{2}z^{6}+2z^{6}a^{-2}-8z^{6}a^{-4}+21z^{6}+11a^{3}z^{5}+34az^{5}+40z^{5}a^{-1}+13z^{5}a^{-3}-4z^{5}a^{-5}+2a^{4}z^{4}-2a^{2}z^{4}+11z^{4}a^{-2}+8z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}-9a^{3}z^{3}-27az^{3}-27z^{3}a^{-1}-7z^{3}a^{-3}+2z^{3}a^{-5}-a^{4}z^{2}-2a^{2}z^{2}-6z^{2}a^{-2}-3z^{2}a^{-4}-4z^{2}+a^{3}z+5az+6za^{-1}+2za^{-3}-a^{2}+a^{3}z^{-1}+az^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

3

8

5

1

-4

6

7

3

4

8

5

-3

2

8

7

1

0

7

9

2

-2

5

7

-2

-4

3

8

5

-6

1

4

-3

-8

3

-10

1

-1

Integral Khovanov Homology

(db, data source)

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