L11a480

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_18,7,19,8 X_14,21,15,22 X_20,10,21,9 X_8,13,9,14 X_22,15,17,16 X_16,17,5,18 X_12,20,13,19 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {8, -3, 9, -5, 4, -7}, {10, -1, 3, -6, 5, -2, 11, -9, 6, -4, 7, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)^{3}(t(2)t(3)+1)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$q^{3}-4q^{2}+9q-13+19q^{-1}-20q^{-2}+21q^{-3}-17q^{-4}+13q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{6}z^{4}-3a^{6}z^{2}-2a^{6}+2a^{4}z^{6}+8a^{4}z^{4}+11a^{4}z^{2}+a^{4}z^{-2}+6a^{4}-a^{2}z^{8}-5a^{2}z^{6}-10a^{2}z^{4}-11a^{2}z^{2}-2a^{2}z^{-2}-7a^{2}+z^{6}+3z^{4}+3z^{2}+z^{-2}+3$ (db)
Kauffman polynomial	$2a^{4}z^{10}+2a^{2}z^{10}+5a^{5}z^{9}+12a^{3}z^{9}+7az^{9}+6a^{6}z^{8}+10a^{4}z^{8}+12a^{2}z^{8}+8z^{8}+5a^{7}z^{7}-2a^{5}z^{7}-26a^{3}z^{7}-15az^{7}+4z^{7}a^{-1}+3a^{8}z^{6}-6a^{6}z^{6}-32a^{4}z^{6}-47a^{2}z^{6}+z^{6}a^{-2}-23z^{6}+a^{9}z^{5}-6a^{7}z^{5}-5a^{5}z^{5}+18a^{3}z^{5}+7az^{5}-9z^{5}a^{-1}-5a^{8}z^{4}+a^{6}z^{4}+39a^{4}z^{4}+56a^{2}z^{4}-2z^{4}a^{-2}+21z^{4}-2a^{9}z^{3}+a^{7}z^{3}+6a^{5}z^{3}+a^{3}z^{3}+az^{3}+3z^{3}a^{-1}+3a^{8}z^{2}-23a^{4}z^{2}-30a^{2}z^{2}-10z^{2}+a^{9}z+a^{7}z-3a^{5}z-6a^{3}z-3az-a^{8}+7a^{4}+9a^{2}+4+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

3

-3

3

6

1

5

1

7

3

-4

-1

12

6

-3

10

9

-1

-5

11

10

1

-7

8

12

4

-9

5

9

-4

-11

2

8

6

-13

1

5

-4

-15

2

-17

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-3$	$i=-1$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-3$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{11}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{12}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$