L11a433

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,6,15,5 X₈₄₉₃ X_2,16,3,15 X_16,7,17,8 X_18,14,19,13 X_22,9,13,10 X_20,11,21,12 X_12,19,5,20 X_10,21,11,22 X_4,17,1,18
Gauss code	{1, -4, 3, -11}, {2, -1, 5, -3, 7, -10, 8, -9}, {6, -2, 4, -5, 11, -6, 9, -8, 10, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)\left(t(3)t(2)^{2}-t(2)^{2}+t(3)^{2}t(2)-t(3)t(2)+t(2)-t(3)^{2}+t(3)\right)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-7}+4q^{-6}-6q^{-5}+q^{4}+11q^{-4}-4q^{3}-14q^{-3}+8q^{2}+18q^{-2}-12q-17q^{-1}+16$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{6}+a^{6}z^{-2}+2z^{4}a^{4}+3z^{2}a^{4}-2a^{4}z^{-2}-a^{4}-z^{6}a^{2}-2z^{4}a^{2}-2z^{2}a^{2}+a^{2}z^{-2}-z^{6}-2z^{4}-z^{2}+1+z^{4}a^{-2}+z^{2}a^{-2}$ (db)
Kauffman polynomial	$a^{7}z^{7}-3a^{7}z^{5}+a^{7}z^{3}+4a^{6}z^{8}-16a^{6}z^{6}+17a^{6}z^{4}-3a^{6}z^{2}+a^{6}z^{-2}-3a^{6}+5a^{5}z^{9}-18a^{5}z^{7}+17a^{5}z^{5}-5a^{5}z^{3}+4a^{5}z-2a^{5}z^{-1}+2a^{4}z^{10}+4a^{4}z^{8}-35a^{4}z^{6}+45a^{4}z^{4}+z^{4}a^{-4}-13a^{4}z^{2}+2a^{4}z^{-2}-4a^{4}+11a^{3}z^{9}-34a^{3}z^{7}+30a^{3}z^{5}+4z^{5}a^{-3}-9a^{3}z^{3}-2z^{3}a^{-3}+4a^{3}z-2a^{3}z^{-1}+2a^{2}z^{10}+9a^{2}z^{8}-38a^{2}z^{6}+8z^{6}a^{-2}+39a^{2}z^{4}-8z^{4}a^{-2}-14a^{2}z^{2}+2z^{2}a^{-2}+a^{2}z^{-2}-a^{2}+6az^{9}-5az^{7}+10z^{7}a^{-1}-6az^{5}-12z^{5}a^{-1}+2az^{3}+3z^{3}a^{-1}+9z^{8}-11z^{6}+2z^{4}-2z^{2}+1$ (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

9

1

7

3

-3

5

1

4

3

7

3

-4

1

9

5

4

-1

10

9

-1

-3

8

7

1

-5

6

10

4

-7

5

8

-3

-9

3

8

5

-11

1

3

-2

-13

3

-15

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{5}$
$r=-3$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$