L11a460

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,4,15,3 X_22,10,13,9 X_20,8,21,7 X_8,14,9,13 X_18,15,19,16 X_16,6,17,5 X_12,18,5,17 X_10,22,11,21 X_2,11,3,12 X_4,20,1,19
Gauss code	{1, -10, 2, -11}, {7, -1, 4, -5, 3, -9, 10, -8}, {5, -2, 6, -7, 8, -6, 11, -4, 9, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(3)^{2}t(2)^{3}-t(3)^{2}t(2)^{3}-2t(1)t(3)t(2)^{3}+2t(3)t(2)^{3}-t(2)^{3}+t(1)t(3)^{3}t(2)^{2}-t(3)^{3}t(2)^{2}-5t(1)t(3)^{2}t(2)^{2}+5t(3)^{2}t(2)^{2}-2t(1)t(2)^{2}+6t(1)t(3)t(2)^{2}-6t(3)t(2)^{2}+2t(2)^{2}-2t(1)t(3)^{3}t(2)+2t(3)^{3}t(2)+6t(1)t(3)^{2}t(2)-6t(3)^{2}t(2)+t(1)t(2)-5t(1)t(3)t(2)+5t(3)t(2)-t(2)+t(1)t(3)^{3}-2t(1)t(3)^{2}+2t(3)^{2}+t(1)t(3)-t(3)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{9}-3q^{8}+8q^{7}-12q^{6}+19q^{5}-22q^{4}+23q^{3}-20q^{2}-q^{-2}+16q+5q^{-1}-10$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+z^{4}a^{-2}+z^{4}a^{-4}-2z^{4}a^{-6}-z^{4}+z^{2}a^{-4}-3z^{2}a^{-6}+z^{2}a^{-8}+a^{-4}-3a^{-6}+a^{-8}+1+a^{-4}z^{-2}-2a^{-6}z^{-2}+a^{-8}z^{-2}$ (db)
Kauffman polynomial	$z^{6}a^{-10}-3z^{4}a^{-10}+2z^{2}a^{-10}+3z^{7}a^{-9}-7z^{5}a^{-9}+3z^{3}a^{-9}+6z^{8}a^{-8}-18z^{6}a^{-8}+24z^{4}a^{-8}-20z^{2}a^{-8}-a^{-8}z^{-2}+8a^{-8}+5z^{9}a^{-7}-6z^{7}a^{-7}-7z^{5}a^{-7}+17z^{3}a^{-7}-11za^{-7}+2a^{-7}z^{-1}+2z^{10}a^{-6}+10z^{8}a^{-6}-36z^{6}a^{-6}+46z^{4}a^{-6}-31z^{2}a^{-6}-2a^{-6}z^{-2}+13a^{-6}+12z^{9}a^{-5}-19z^{7}a^{-5}+z^{5}a^{-5}+17z^{3}a^{-5}-11za^{-5}+2a^{-5}z^{-1}+2z^{10}a^{-4}+15z^{8}a^{-4}-36z^{6}a^{-4}+26z^{4}a^{-4}-10z^{2}a^{-4}-a^{-4}z^{-2}+5a^{-4}+7z^{9}a^{-3}-14z^{5}a^{-3}+6z^{3}a^{-3}+11z^{8}a^{-2}-14z^{6}a^{-2}+2z^{4}a^{-2}-z^{2}a^{-2}+10z^{7}a^{-1}+az^{5}-14z^{5}a^{-1}+3z^{3}a^{-1}+5z^{6}-5z^{4}+1$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

7

8

χ

19

1

17

3

1

-2

15

5

13

7

3

-4

11

12

5

7

9

11

8

-3

7

12

11

1

5

8

11

3

8

12

-4

1

4

10

6

-1

1

6

-5

-3

4

-5

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=3$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=4$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{12}$
$r=5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=7$	${\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=8$	${\mathbb {Z} }$	${\mathbb {Z} }$