L11a533

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Link Presentations

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

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

Polynomial invariants

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

10

1

-1

8

3

6

4

1

-3

4

5

3

2

6

4

-2

0

8

5

3

-2

7

10

3

-4

4

0

-6

1

7

6

-8

4

0

-10

1

5

4

-12

0

-14

1

Integral Khovanov Homology

(db, data source)

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