L11a275

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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_14,5,15,6 X_6,9,7,10 X_18,8,19,7 X_22,16,9,15 X_20,18,21,17 X_16,22,17,21 X_8,20,1,19 X_4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -5, 6, -10}, {5, -1, 2, -3, 11, -4, 7, -9, 8, -6, 10, -8, 9, -7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(2)^{3}t(1)^{3}-2t(2)^{2}t(1)^{3}-2t(2)^{3}t(1)^{2}+5t(2)^{2}t(1)^{2}-2t(2)t(1)^{2}-2t(2)^{2}t(1)+5t(2)t(1)-2t(1)-2t(2)+2}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{7/2}-2q^{5/2}+3q^{3/2}-5{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {7}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {6}{q^{9/2}}}-{\frac {4}{q^{11/2}}}+{\frac {2}{q^{13/2}}}-{\frac {1}{q^{15/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{5}z^{5}+4a^{5}z^{3}+4a^{5}z+a^{5}z^{-1}-a^{3}z^{7}-5a^{3}z^{5}-8a^{3}z^{3}-6a^{3}z-a^{3}z^{-1}-az^{7}-5az^{5}+z^{5}a^{-1}-7az^{3}+4z^{3}a^{-1}-3az+3za^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-2a^{3}z^{9}-4az^{9}-2z^{9}a^{-1}-3a^{4}z^{8}+a^{2}z^{8}-z^{8}a^{-2}+3z^{8}-4a^{5}z^{7}+3a^{3}z^{7}+19az^{7}+12z^{7}a^{-1}-4a^{6}z^{6}+5a^{4}z^{6}+6a^{2}z^{6}+6z^{6}a^{-2}+3z^{6}-3a^{7}z^{5}+8a^{5}z^{5}+7a^{3}z^{5}-27az^{5}-23z^{5}a^{-1}-2a^{8}z^{4}+6a^{6}z^{4}+3a^{4}z^{4}-6a^{2}z^{4}-11z^{4}a^{-2}-12z^{4}-a^{9}z^{3}+2a^{7}z^{3}-6a^{5}z^{3}-11a^{3}z^{3}+14az^{3}+16z^{3}a^{-1}+a^{8}z^{2}-4a^{6}z^{2}-4a^{4}z^{2}+a^{2}z^{2}+6z^{2}a^{-2}+6z^{2}+a^{9}z-a^{7}z+4a^{5}z+7a^{3}z-3az-4za^{-1}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

8

1

-1

6

1

4

2

1

-1

2

3

1

2

0

3

2

-1

-2

5

3

2

-4

3

4

1

-6

4

0

-8

2

3

1

-10

2

4

-2

-12

1

3

2

-14

1

-1

-16

1

Integral Khovanov Homology

(db, data source)

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