L11a98

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(1)t(2)^{5}-2t(2)^{5}-5t(1)t(2)^{4}+7t(2)^{4}+8t(1)t(2)^{3}-9t(2)^{3}-9t(1)t(2)^{2}+8t(2)^{2}+7t(1)t(2)-5t(2)-2t(1)+2}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+10q^{7/2}-15q^{5/2}+19q^{3/2}-22{\sqrt {q}}+{\frac {20}{\sqrt {q}}}-{\frac {18}{q^{3/2}}}+{\frac {12}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-az^{7}-z^{7}a^{-1}+a^{3}z^{5}-4az^{5}-3z^{5}a^{-1}+z^{5}a^{-3}+3a^{3}z^{3}-7az^{3}-3z^{3}a^{-1}+2z^{3}a^{-3}+3a^{3}z-5az-2za^{-1}+2za^{-3}+a^{3}z^{-1}-2a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}+a^{5}z^{7}-4a^{5}z^{5}+4z^{5}a^{-5}+5a^{5}z^{3}-2a^{5}z+3a^{4}z^{8}-11a^{4}z^{6}+10z^{6}a^{-4}+13a^{4}z^{4}-9z^{4}a^{-4}-5a^{4}z^{2}+5z^{2}a^{-4}-2a^{-4}+4a^{3}z^{9}-11a^{3}z^{7}+15z^{7}a^{-3}+7a^{3}z^{5}-21z^{5}a^{-3}+10z^{3}a^{-3}+2a^{3}z-2za^{-3}-a^{3}z^{-1}+a^{-3}z^{-1}+2a^{2}z^{10}+5a^{2}z^{8}+14z^{8}a^{-2}-30a^{2}z^{6}-18z^{6}a^{-2}+34a^{2}z^{4}-2z^{4}a^{-2}-12a^{2}z^{2}+10z^{2}a^{-2}+a^{2}-5a^{-2}+12az^{9}+8z^{9}a^{-1}-28az^{7}-z^{7}a^{-1}+14az^{5}-22z^{5}a^{-1}-2az^{3}+13z^{3}a^{-1}+2az-4za^{-1}+2a^{-1}z^{-1}+2z^{10}+16z^{8}-47z^{6}+29z^{4}-2z^{2}-3$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

3

8

7

1

-6

6

8

3

5

4

11

7

-4

2

11

8

3

0

10

12

2

-2

8

10

-2

-4

4

10

6

-6

3

8

-5

-8

1

5

4

-10

2

-2

-12

1

Integral Khovanov Homology

(db, data source)

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