L10a60

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{4}-3u^{2}v^{3}+4u^{2}v^{2}-2u^{2}v-uv^{4}+4uv^{3}-7uv^{2}+4uv-u-2v^{3}+4v^{2}-3v+1}{uv^{2}}}$ (db)
Jones polynomial	$q^{11/2}-3q^{9/2}+6q^{7/2}-10q^{5/2}+11q^{3/2}-13{\sqrt {q}}+{\frac {11}{\sqrt {q}}}-{\frac {9}{q^{3/2}}}+{\frac {6}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-3}-a^{3}z^{3}+3z^{3}a^{-3}-2a^{3}z+3za^{-3}-a^{3}z^{-1}-z^{7}a^{-1}+2az^{5}-5z^{5}a^{-1}+7az^{3}-10z^{3}a^{-1}+8az-8za^{-1}+3az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}-z^{2}a^{-6}+3z^{5}a^{-5}-3z^{3}a^{-5}+za^{-5}+a^{4}z^{6}+5z^{6}a^{-4}-3a^{4}z^{4}-5z^{4}a^{-4}+3a^{4}z^{2}+2z^{2}a^{-4}-a^{4}+3a^{3}z^{7}+6z^{7}a^{-3}-9a^{3}z^{5}-8z^{5}a^{-3}+8a^{3}z^{3}+7z^{3}a^{-3}-3a^{3}z-3za^{-3}+a^{3}z^{-1}+3a^{2}z^{8}+4z^{8}a^{-2}-4a^{2}z^{6}-7a^{2}z^{4}-8z^{4}a^{-2}+10a^{2}z^{2}+6z^{2}a^{-2}-3a^{2}+az^{9}+z^{9}a^{-1}+8az^{7}+11z^{7}a^{-1}-29az^{5}-31z^{5}a^{-1}+28az^{3}+30z^{3}a^{-1}-13az-14za^{-1}+3az^{-1}+2a^{-1}z^{-1}+7z^{8}-10z^{6}-6z^{4}+10z^{2}-3$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

2

8

4

1

-3

6

2

4

6

5

-1

2

7

5

2

0

5

7

2

-2

4

6

-2

-4

2

5

3

-6

1

4

-3

-8

2

-10

1

-1

Integral Khovanov Homology

(db, data source)

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