L10a30

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_14,8,15,7 X_18,10,19,9 X_20,16,5,15 X_16,20,17,19 X_8,18,9,17 X_10,14,11,13 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -7, 4, -8, 10, -2, 8, -3, 5, -6, 7, -4, 6, -5}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

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

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

3

-3

10

4

1

3

8

6

3

-3

6

4

2

4

7

6

-1

2

5

6

-1

0

4

8

4

-2

2

4

-2

-4

4

-6

1

2

-1

-8

1

Integral Khovanov Homology

(db, data source)

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