L11a325

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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_18,14,19,13 X_14,8,15,7 X_20,5,21,6 X_4,19,5,20 X_6,21,7,22 X_16,9,17,10 X_22,15,9,16 X_8,18,1,17
Gauss code	{1, -2, 3, -7, 6, -8, 5, -11}, {9, -1, 2, -3, 4, -5, 10, -9, 11, -4, 7, -6, 8, -10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{3}t(2)^{5}-t(1)^{2}t(2)^{5}-2t(1)^{3}t(2)^{4}+4t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}+2t(1)^{3}t(2)^{3}-4t(1)^{2}t(2)^{3}+4t(1)t(2)^{3}-t(2)^{3}-t(1)^{3}t(2)^{2}+4t(1)^{2}t(2)^{2}-4t(1)t(2)^{2}+2t(2)^{2}-2t(1)^{2}t(2)+4t(1)t(2)-2t(2)-t(1)+1}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	${\sqrt {q}}-{\frac {3}{\sqrt {q}}}+{\frac {5}{q^{3/2}}}-{\frac {9}{q^{5/2}}}+{\frac {10}{q^{7/2}}}-{\frac {13}{q^{9/2}}}+{\frac {13}{q^{11/2}}}-{\frac {11}{q^{13/2}}}+{\frac {9}{q^{15/2}}}-{\frac {6}{q^{17/2}}}+{\frac {3}{q^{19/2}}}-{\frac {1}{q^{21/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{7}z^{7}+5a^{7}z^{5}+8a^{7}z^{3}+5a^{7}z-a^{5}z^{9}-7a^{5}z^{7}-18a^{5}z^{5}-21a^{5}z^{3}-9a^{5}z+a^{5}z^{-1}+a^{3}z^{7}+5a^{3}z^{5}+7a^{3}z^{3}+2a^{3}z-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{13}-3z^{4}a^{12}-6z^{5}a^{11}+4z^{3}a^{11}-za^{11}-9z^{6}a^{10}+13z^{4}a^{10}-5z^{2}a^{10}-10z^{7}a^{9}+19z^{5}a^{9}-7z^{3}a^{9}+za^{9}-9z^{8}a^{8}+21z^{6}a^{8}-10z^{4}a^{8}+4z^{2}a^{8}-6z^{9}a^{7}+14z^{7}a^{7}-2z^{5}a^{7}-z^{3}a^{7}-2za^{7}-2z^{10}a^{6}-3z^{8}a^{6}+33z^{6}a^{6}-43z^{4}a^{6}+16z^{2}a^{6}-9z^{9}a^{5}+40z^{7}a^{5}-55z^{5}a^{5}+29z^{3}a^{5}-7za^{5}-a^{5}z^{-1}-2z^{10}a^{4}+5z^{8}a^{4}+8z^{6}a^{4}-25z^{4}a^{4}+11z^{2}a^{4}+a^{4}-3z^{9}a^{3}+16z^{7}a^{3}-28z^{5}a^{3}+18z^{3}a^{3}-3za^{3}-a^{3}z^{-1}-z^{8}a^{2}+5z^{6}a^{2}-8z^{4}a^{2}+4z^{2}a^{2}$ (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

3

χ

2

1

-1

0

2

-2

3

1

-2

-4

6

2

4

-6

5

4

-1

-8

8

5

3

-10

5

0

-12

6

8

-2

-14

4

6

2

-16

2

5

-3

-18

1

4

3

-20

2

-2

-22

1

Integral Khovanov Homology

(db, data source)

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