L10a25

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Link Presentations

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2uv^{4}-6uv^{3}+6uv^{2}-4uv+u+v^{5}-4v^{4}+6v^{3}-6v^{2}+2v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	${\frac {13}{q^{9/2}}}-{\frac {12}{q^{7/2}}}+{\frac {9}{q^{5/2}}}-{\frac {7}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {6}{q^{15/2}}}+{\frac {9}{q^{13/2}}}-{\frac {12}{q^{11/2}}}-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$za^{9}+a^{9}z^{-1}-3z^{3}a^{7}-6za^{7}-3a^{7}z^{-1}+2z^{5}a^{5}+6z^{3}a^{5}+8za^{5}+4a^{5}z^{-1}+z^{5}a^{3}-4za^{3}-2a^{3}z^{-1}-z^{3}a-za$ (db)
Kauffman polynomial	$-z^{5}a^{11}+2z^{3}a^{11}-za^{11}-3z^{6}a^{10}+6z^{4}a^{10}-3z^{2}a^{10}+a^{10}-4z^{7}a^{9}+6z^{5}a^{9}-2z^{3}a^{9}+2za^{9}-a^{9}z^{-1}-3z^{8}a^{8}-z^{6}a^{8}+11z^{4}a^{8}-10z^{2}a^{8}+3a^{8}-z^{9}a^{7}-9z^{7}a^{7}+25z^{5}a^{7}-27z^{3}a^{7}+13za^{7}-3a^{7}z^{-1}-7z^{8}a^{6}+9z^{6}a^{6}+2z^{4}a^{6}-10z^{2}a^{6}+3a^{6}-z^{9}a^{5}-10z^{7}a^{5}+29z^{5}a^{5}-33z^{3}a^{5}+18za^{5}-4a^{5}z^{-1}-4z^{8}a^{4}+4z^{6}a^{4}+2z^{4}a^{4}-4z^{2}a^{4}+2a^{4}-5z^{7}a^{3}+10z^{5}a^{3}-8z^{3}a^{3}+7za^{3}-2a^{3}z^{-1}-3z^{6}a^{2}+5z^{4}a^{2}-z^{2}a^{2}-z^{5}a+2z^{3}a-za$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

2

-2

5

1

4

-4

5

3

-2

-6

7

4

3

-8

6

5

-1

-10

6

7

-1

-12

4

7

3

-14

2

5

-3

-16

1

4

3

-18

2

-2

-20

1

Integral Khovanov Homology

(db, data source)

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