L11a203

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X₆₇₁₈ X_18,11,19,12 X_16,6,17,5 X_4,18,5,17 X_22,15,7,16 X_12,21,13,22 X_20,13,21,14 X_14,19,15,20
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 5, -9, 10, -11, 8, -6, 7, -5, 11, -10, 9, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2u^{2}v^{4}-3u^{2}v^{3}+3u^{2}v^{2}-2u^{2}v-2uv^{4}+3uv^{3}-3uv^{2}+3uv-2u-2v^{3}+3v^{2}-3v+2}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {5}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {9}{q^{9/2}}}+{\frac {10}{q^{11/2}}}-{\frac {10}{q^{13/2}}}+{\frac {9}{q^{15/2}}}-{\frac {7}{q^{17/2}}}+{\frac {4}{q^{19/2}}}-{\frac {2}{q^{21/2}}}+{\frac {1}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-z^{5}a^{9}-4z^{3}a^{9}-4za^{9}-a^{9}z^{-1}+z^{7}a^{7}+5z^{5}a^{7}+9z^{3}a^{7}+8za^{7}+2a^{7}z^{-1}+z^{7}a^{5}+4z^{5}a^{5}+3z^{3}a^{5}-za^{5}-z^{5}a^{3}-4z^{3}a^{3}-4za^{3}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{4}a^{14}+2z^{2}a^{14}-2z^{5}a^{13}+3z^{3}a^{13}-3z^{6}a^{12}+5z^{4}a^{12}-3z^{2}a^{12}-4z^{7}a^{11}+10z^{5}a^{11}-12z^{3}a^{11}+2za^{11}-4z^{8}a^{10}+12z^{6}a^{10}-18z^{4}a^{10}+9z^{2}a^{10}-2a^{10}-3z^{9}a^{9}+9z^{7}a^{9}-13z^{5}a^{9}+9z^{3}a^{9}-4za^{9}+a^{9}z^{-1}-z^{10}a^{8}-2z^{8}a^{8}+17z^{6}a^{8}-31z^{4}a^{8}+25z^{2}a^{8}-5a^{8}-5z^{9}a^{7}+20z^{7}a^{7}-31z^{5}a^{7}+29z^{3}a^{7}-12za^{7}+2a^{7}z^{-1}-z^{10}a^{6}+10z^{6}a^{6}-14z^{4}a^{6}+10z^{2}a^{6}-3a^{6}-2z^{9}a^{5}+6z^{7}a^{5}-z^{5}a^{5}-3z^{3}a^{5}-za^{5}-2z^{8}a^{4}+8z^{6}a^{4}-7z^{4}a^{4}-z^{2}a^{4}+a^{4}-z^{7}a^{3}+5z^{5}a^{3}-8z^{3}a^{3}+5za^{3}-a^{3}z^{-1}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

0

1

-2

1

-1

-4

4

1

3

-6

3

2

-1

-8

6

3

-10

5

4

-1

-12

5

0

-14

4

5

1

-16

3

5

-2

-18

1

4

3

-20

1

3

-2

-22

1

-24

1

-1

Integral Khovanov Homology

(db, data source)

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