L11a506

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_16,8,17,7 X_14,6,15,5 X_10,4,11,3 X_4,14,5,13 X_2,18,3,17 X_18,9,19,10 X_12,21,7,22 X_22,11,13,12 X_20,16,21,15 X_6,19,1,20
Gauss code	{1, -6, 4, -5, 3, -11}, {2, -1, 7, -4, 9, -8}, {5, -3, 10, -2, 6, -7, 11, -10, 8, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(3)-1)\left(t(2)^{2}t(1)^{2}+t(2)^{2}t(3)^{2}t(1)^{2}-2t(2)t(3)^{2}t(1)^{2}+t(3)^{2}t(1)^{2}-t(2)t(1)^{2}-t(2)^{2}t(3)t(1)^{2}+2t(2)t(3)t(1)^{2}-t(3)t(1)^{2}-2t(2)^{2}t(1)-t(2)^{2}t(3)^{2}t(1)+3t(2)t(3)^{2}t(1)-2t(3)^{2}t(1)+3t(2)t(1)+2t(2)^{2}t(3)t(1)-3t(2)t(3)t(1)+2t(3)t(1)-t(1)+t(2)^{2}-t(2)t(3)^{2}+t(3)^{2}-2t(2)-t(2)^{2}t(3)+2t(2)t(3)-t(3)+1\right)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{7}-4q^{6}+9q^{5}-16q^{4}+22q^{3}-24q^{2}+26q-21+17q^{-1}-10q^{-2}+5q^{-3}-q^{-4}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-4}+3z^{4}a^{-4}+3z^{2}a^{-4}-z^{8}a^{-2}-5z^{6}a^{-2}-a^{2}z^{4}-10z^{4}a^{-2}-a^{2}z^{2}-7z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+a^{2}+a^{-2}+2z^{6}+6z^{4}+4z^{2}-2z^{-2}-2$ (db)
Kauffman polynomial	$z^{4}a^{-8}+4z^{5}a^{-7}-z^{3}a^{-7}+9z^{6}a^{-6}-6z^{4}a^{-6}+z^{2}a^{-6}+15z^{7}a^{-5}-20z^{5}a^{-5}+9z^{3}a^{-5}+18z^{8}a^{-4}-34z^{6}a^{-4}+24z^{4}a^{-4}-8z^{2}a^{-4}+13z^{9}a^{-3}+a^{3}z^{7}-17z^{7}a^{-3}-2a^{3}z^{5}-8z^{5}a^{-3}+a^{3}z^{3}+10z^{3}a^{-3}+4z^{10}a^{-2}+5a^{2}z^{8}+19z^{8}a^{-2}-15a^{2}z^{6}-77z^{6}a^{-2}+13a^{2}z^{4}+71z^{4}a^{-2}-2a^{2}z^{2}-21z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-2a^{2}-2a^{-2}+8az^{9}+21z^{9}a^{-1}-23az^{7}-56z^{7}a^{-1}+16az^{5}+34z^{5}a^{-1}-az^{3}-2z^{3}a^{-1}+2az+2za^{-1}-2az^{-1}-2a^{-1}z^{-1}+4z^{10}+6z^{8}-49z^{6}+53z^{4}-14z^{2}+2z^{-2}-3$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

15

1

13

3

-3

11

6

1

5

9

10

3

-7

7

12

6

5

13

11

-2

3

13

11

2

1

10

15

5

-1

7

11

-4

-3

4

11

7

-5

1

6

-5

-7

4

-9

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{13}$
$r=1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{12}$
$r=3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$