L11a493

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(3)-1)\left(t(3)^{2}-t(3)+1\right)\left(t(2)t(3)^{2}-2t(2)t(3)+2t(3)-1\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$q^{10}-4q^{9}+10q^{8}-16q^{7}+20q^{6}-23q^{5}+24q^{4}-18q^{3}+15q^{2}-8q+4-q^{-1}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^{8}a^{-4}-z^{6}a^{-2}+5z^{6}a^{-4}-2z^{6}a^{-6}-3z^{4}a^{-2}+10z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}-2z^{2}a^{-2}+9z^{2}a^{-4}-9z^{2}a^{-6}+2z^{2}a^{-8}+a^{-2}+2a^{-4}-5a^{-6}+2a^{-8}+a^{-2}z^{-2}-2a^{-4}z^{-2}+a^{-6}z^{-2}$ (db)
Kauffman polynomial	$2z^{10}a^{-4}+2z^{10}a^{-6}+5z^{9}a^{-3}+14z^{9}a^{-5}+9z^{9}a^{-7}+4z^{8}a^{-2}+12z^{8}a^{-4}+24z^{8}a^{-6}+16z^{8}a^{-8}+z^{7}a^{-1}-11z^{7}a^{-3}-26z^{7}a^{-5}+2z^{7}a^{-7}+16z^{7}a^{-9}-14z^{6}a^{-2}-56z^{6}a^{-4}-76z^{6}a^{-6}-24z^{6}a^{-8}+10z^{6}a^{-10}-3z^{5}a^{-1}-2z^{5}a^{-3}-11z^{5}a^{-5}-39z^{5}a^{-7}-23z^{5}a^{-9}+4z^{5}a^{-11}+17z^{4}a^{-2}+66z^{4}a^{-4}+66z^{4}a^{-6}+8z^{4}a^{-8}-8z^{4}a^{-10}+z^{4}a^{-12}+3z^{3}a^{-1}+14z^{3}a^{-3}+32z^{3}a^{-5}+34z^{3}a^{-7}+13z^{3}a^{-9}-8z^{2}a^{-2}-29z^{2}a^{-4}-25z^{2}a^{-6}+4z^{2}a^{-10}-za^{-1}-3za^{-3}-11za^{-5}-13za^{-7}-4za^{-9}+4a^{-4}+5a^{-6}+a^{-8}-a^{-10}-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-2}z^{-2}+2a^{-4}z^{-2}+a^{-6}z^{-2}$ (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

21

1

19

3

-3

17

7

1

6

15

9

3

-6

13

11

7

4

11

14

11

-3

9

10

9

1

7

8

14

6

5

7

10

-3

3

10

7

1

5

-4

-1

3

-3

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=3$	$i=5$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{7}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=4$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{11}$
$r=5$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=7$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=8$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$