L11a253

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{3}t(2)^{5}-t(1)^{2}t(2)^{5}-2t(1)^{3}t(2)^{4}+3t(1)^{2}t(2)^{4}-t(1)t(2)^{4}+2t(1)^{3}t(2)^{3}-5t(1)^{2}t(2)^{3}+3t(1)t(2)^{3}-t(2)^{3}-t(1)^{3}t(2)^{2}+3t(1)^{2}t(2)^{2}-5t(1)t(2)^{2}+2t(2)^{2}-t(1)^{2}t(2)+3t(1)t(2)-2t(2)-t(1)+1}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	${\frac {10}{q^{9/2}}}-{\frac {12}{q^{7/2}}}-q^{5/2}+{\frac {11}{q^{5/2}}}+3q^{3/2}-{\frac {10}{q^{3/2}}}+{\frac {1}{q^{17/2}}}-{\frac {3}{q^{15/2}}}+{\frac {5}{q^{13/2}}}-{\frac {8}{q^{11/2}}}-5{\sqrt {q}}+{\frac {7}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{3}z^{9}-a^{5}z^{7}+7a^{3}z^{7}-az^{7}-5a^{5}z^{5}+17a^{3}z^{5}-5az^{5}-7a^{5}z^{3}+16a^{3}z^{3}-7az^{3}-2a^{5}z+3a^{3}z-3az+a^{5}z^{-1}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{4}a^{10}+z^{2}a^{10}-3z^{5}a^{9}+4z^{3}a^{9}-za^{9}-4z^{6}a^{8}+4z^{4}a^{8}-4z^{7}a^{7}+3z^{5}a^{7}+2z^{3}a^{7}-2za^{7}-4z^{8}a^{6}+6z^{6}a^{6}-3z^{4}a^{6}-4z^{9}a^{5}+12z^{7}a^{5}-17z^{5}a^{5}+11z^{3}a^{5}-za^{5}-a^{5}z^{-1}-2z^{10}a^{4}+3z^{8}a^{4}+5z^{6}a^{4}-10z^{4}a^{4}+3z^{2}a^{4}+a^{4}-8z^{9}a^{3}+35z^{7}a^{3}-51z^{5}a^{3}+30z^{3}a^{3}-4za^{3}-a^{3}z^{-1}-2z^{10}a^{2}+4z^{8}a^{2}+8z^{6}a^{2}-16z^{4}a^{2}+6z^{2}a^{2}-4z^{9}a+18z^{7}a-24z^{5}a+14z^{3}a-4za-3z^{8}+13z^{6}-14z^{4}+4z^{2}-z^{7}a^{-1}+4z^{5}a^{-1}-3z^{3}a^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

6

1

4

2

-2

2

3

1

2

0

4

2

-2

6

3

-4

6

5

-1

-6

6

5

1

-8

4

6

2

-10

4

6

-2

-12

2

5

3

-14

1

3

-2

-16

2

-18

1

-1

Integral Khovanov Homology

(db, data source)

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