L10a115

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X_20,9,11,10 X_14,3,15,4 X_16,5,17,6 X_4,15,5,16 X_8,17,9,18 X_18,7,19,8 X_6,19,7,20 X_2,11,3,12 X_10,13,1,14
Gauss code	{1, -9, 3, -5, 4, -8, 7, -6, 2, -10}, {9, -1, 10, -3, 5, -4, 6, -7, 8, -2}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{4}\left(-v^{2}\right)-2u^{3}v^{3}+2u^{3}v^{2}-u^{3}v-u^{2}v^{4}+2u^{2}v^{3}-3u^{2}v^{2}+2u^{2}v-u^{2}-uv^{3}+2uv^{2}-2uv-v^{2}}{u^{2}v^{2}}}$ (db)
Jones polynomial	$-{\frac {4}{q^{9/2}}}+{\frac {2}{q^{7/2}}}-{\frac {1}{q^{5/2}}}-{\frac {1}{q^{25/2}}}+{\frac {1}{q^{23/2}}}-{\frac {3}{q^{21/2}}}+{\frac {5}{q^{19/2}}}-{\frac {6}{q^{17/2}}}+{\frac {7}{q^{15/2}}}-{\frac {7}{q^{13/2}}}+{\frac {5}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z^{3}a^{11}+3za^{11}+a^{11}z^{-1}-z^{5}a^{9}-3z^{3}a^{9}-2za^{9}-a^{9}z^{-1}-2z^{5}a^{7}-7z^{3}a^{7}-5za^{7}-z^{5}a^{5}-3z^{3}a^{5}-za^{5}$ (db)
Kauffman polynomial	$-z^{5}a^{15}+4z^{3}a^{15}-4za^{15}-z^{6}a^{14}+2z^{4}a^{14}-z^{7}a^{13}+z^{5}a^{13}+z^{3}a^{13}-z^{8}a^{12}+2z^{6}a^{12}-5z^{4}a^{12}+6z^{2}a^{12}-z^{9}a^{11}+4z^{7}a^{11}-12z^{5}a^{11}+13z^{3}a^{11}-6za^{11}+a^{11}z^{-1}-3z^{8}a^{10}+10z^{6}a^{10}-16z^{4}a^{10}+7z^{2}a^{10}-a^{10}-z^{9}a^{9}+2z^{7}a^{9}-3z^{5}a^{9}+2z^{3}a^{9}-4za^{9}+a^{9}z^{-1}-2z^{8}a^{8}+5z^{6}a^{8}-4z^{4}a^{8}-3z^{7}a^{7}+10z^{5}a^{7}-11z^{3}a^{7}+5za^{7}-2z^{6}a^{6}+5z^{4}a^{6}-z^{2}a^{6}-z^{5}a^{5}+3z^{3}a^{5}-za^{5}$ (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		\

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

2

1

-1

-8

2

-10

3

2

-1

-12

4

2

-14

3

0

-16

3

4

-1

-18

2

3

1

-20

1

3

-2

-22

2

-24

1

0

-26

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-6$	$i=-4$
$r=-10$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-9$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-5$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-1$	${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$