L11a205

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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_18,15,19,16 X_16,6,17,5 X_4,18,5,17 X_20,11,21,12 X_22,13,7,14 X_12,21,13,22 X_14,19,15,20
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 8, -10, 9, -11, 5, -6, 7, -5, 11, -8, 10, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}-3t(1)^{2}t(2)^{3}+4t(1)t(2)^{3}-2t(2)^{3}+3t(1)^{2}t(2)^{2}-5t(1)t(2)^{2}+3t(2)^{2}-2t(1)^{2}t(2)+4t(1)t(2)-3t(2)-2t(1)+2}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {9}{q^{9/2}}}+{\frac {6}{q^{7/2}}}-{\frac {4}{q^{5/2}}}+{\frac {1}{q^{3/2}}}+{\frac {1}{q^{23/2}}}-{\frac {3}{q^{21/2}}}+{\frac {6}{q^{19/2}}}-{\frac {9}{q^{17/2}}}+{\frac {11}{q^{15/2}}}-{\frac {12}{q^{13/2}}}+{\frac {11}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-z^{5}a^{9}-3z^{3}a^{9}-2za^{9}+z^{7}a^{7}+4z^{5}a^{7}+4z^{3}a^{7}-a^{7}z^{-1}+z^{7}a^{5}+5z^{5}a^{5}+9z^{3}a^{5}+8za^{5}+3a^{5}z^{-1}-z^{5}a^{3}-5z^{3}a^{3}-7za^{3}-2a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{4}a^{14}+z^{2}a^{14}-3z^{5}a^{13}+3z^{3}a^{13}-5z^{6}a^{12}+6z^{4}a^{12}-2z^{2}a^{12}-6z^{7}a^{11}+10z^{5}a^{11}-8z^{3}a^{11}+za^{11}-5z^{8}a^{10}+9z^{6}a^{10}-9z^{4}a^{10}+3z^{2}a^{10}-3z^{9}a^{9}+5z^{7}a^{9}-7z^{5}a^{9}+8z^{3}a^{9}-3za^{9}-z^{10}a^{8}-z^{8}a^{8}+3z^{6}a^{8}+2z^{4}a^{8}-2z^{2}a^{8}+a^{8}-4z^{9}a^{7}+13z^{7}a^{7}-18z^{5}a^{7}+15z^{3}a^{7}-2za^{7}-a^{7}z^{-1}-z^{10}a^{6}+3z^{8}a^{6}-8z^{6}a^{6}+19z^{4}a^{6}-15z^{2}a^{6}+3a^{6}-z^{9}a^{5}+z^{7}a^{5}+8z^{5}a^{5}-16z^{3}a^{5}+11za^{5}-3a^{5}z^{-1}-z^{8}a^{4}+3z^{6}a^{4}+z^{4}a^{4}-7z^{2}a^{4}+3a^{4}-z^{7}a^{3}+6z^{5}a^{3}-12z^{3}a^{3}+9za^{3}-2a^{3}z^{-1}$ (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

0

-4

4

1

3

-6

3

1

-2

-8

6

3

-10

6

4

-2

-12

6

5

1

-14

5

6

1

-16

4

6

-2

-18

2

5

3

-20

1

4

-3

-22

2

-24

1

-1

Integral Khovanov Homology

(db, data source)

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