L10a96

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Link Presentations

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

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

Polynomial invariants

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

3

-10

4

2

-2

-12

6

3

-14

5

4

-1

-16

5

6

-1

-18

3

5

2

-20

3

5

-2

-22

1

4

3

-24

2

-2

-26

1

Integral Khovanov Homology

(db, data source)

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