L10a45

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Link Presentations

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

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

Polynomial invariants

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

2

-2

10

3

1

2

8

6

2

-4

6

5

3

2

4

6

0

2

5

0

4

7

3

-2

2

4

-2

-4

4

-6

1

2

-1

-8

1

Integral Khovanov Homology

(db, data source)

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