L11a389

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {-t(1)t(3)^{3}+t(1)t(2)t(3)^{3}-2t(2)t(3)^{3}+t(3)^{3}+6t(1)t(3)^{2}-5t(1)t(2)t(3)^{2}+8t(2)t(3)^{2}-5t(3)^{2}-8t(1)t(3)+5t(1)t(2)t(3)-6t(2)t(3)+5t(3)+2t(1)-t(1)t(2)+t(2)-1}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{5}+4q^{4}-8q^{3}+14q^{2}-17q+19-18q^{-1}+16q^{-2}-10q^{-3}+6q^{-4}-2q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-3z^{2}a^{4}-3a^{4}z^{-2}-5a^{4}+3z^{4}a^{2}+6z^{2}a^{2}+4a^{2}z^{-2}+8a^{2}-z^{6}-2z^{4}-5z^{2}-3z^{-2}-6+2z^{4}a^{-2}+2z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}-z^{2}a^{-4}$ (db)
Kauffman polynomial	$a^{2}z^{10}+z^{10}+2a^{3}z^{9}+6az^{9}+4z^{9}a^{-1}+2a^{4}z^{8}+5a^{2}z^{8}+7z^{8}a^{-2}+10z^{8}+2a^{5}z^{7}+3a^{3}z^{7}-az^{7}+5z^{7}a^{-1}+7z^{7}a^{-3}+a^{6}z^{6}+2a^{4}z^{6}-a^{2}z^{6}-5z^{6}a^{-2}+4z^{6}a^{-4}-11z^{6}-5a^{5}z^{5}-12a^{3}z^{5}-11az^{5}-15z^{5}a^{-1}-10z^{5}a^{-3}+z^{5}a^{-5}-4a^{6}z^{4}-18a^{4}z^{4}-26a^{2}z^{4}-7z^{4}a^{-2}-6z^{4}a^{-4}-13z^{4}+3a^{5}z^{3}+8a^{3}z^{3}+5az^{3}+5z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}+6a^{6}z^{2}+25a^{4}z^{2}+38a^{2}z^{2}+8z^{2}a^{-2}+3z^{2}a^{-4}+24z^{2}+a^{5}z+a^{3}z+az+za^{-1}-4a^{6}-14a^{4}-21a^{2}-4a^{-2}-14-a^{5}z^{-1}-a^{3}z^{-1}-az^{-1}-a^{-1}z^{-1}+a^{6}z^{-2}+3a^{4}z^{-2}+4a^{2}z^{-2}+a^{-2}z^{-2}+3z^{-2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

3

7

5

1

-4

5

9

3

6

3

8

5

-3

1

11

9

2

-1

10

11

1

-3

6

8

-2

-5

4

10

6

-7

2

6

-4

-9

1

5

4

-11

1

-1

-13

1

Integral Khovanov Homology

(db, data source)

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