L11n455

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,5,15,6 X_12,4,13,3 X_2,9,3,10 X_7,19,8,18 X_17,9,18,8 X_10,13,5,14 X_19,22,20,17 X_21,11,22,16 X_11,21,12,20 X_4,16,1,15
Gauss code	{1, -4, 3, -11}, {2, -1, -5, 6, 4, -7}, {-10, -3, 7, -2, 11, 9}, {-6, 5, -8, 10, -9, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(1)t(3)^{2}t(2)^{2}+t(3)^{2}t(2)^{2}-t(1)t(2)^{2}+2t(1)t(3)t(2)^{2}+t(1)t(4)t(2)^{2}-t(1)t(3)t(4)t(2)^{2}-2t(3)^{2}t(2)+t(1)t(2)-t(1)t(3)t(2)+t(3)t(2)+t(3)^{2}t(4)t(2)-2t(1)t(4)t(2)+t(1)t(3)t(4)t(2)-t(3)t(4)t(2)+t(3)^{2}-t(3)-t(3)^{2}t(4)+t(1)t(4)+2t(3)t(4)-t(4)}{{\sqrt {t(1)}}t(2)t(3){\sqrt {t(4)}}}}$ (db)
Jones polynomial	$2q^{9/2}-6q^{7/2}+{\frac {1}{q^{7/2}}}+5q^{5/2}-{\frac {4}{q^{5/2}}}-9q^{3/2}+{\frac {5}{q^{3/2}}}-q^{11/2}+7{\sqrt {q}}-{\frac {8}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-2z^{5}a^{-1}+3az^{3}-8z^{3}a^{-1}+3z^{3}a^{-3}-a^{3}z+6az-12za^{-1}+8za^{-3}-za^{-5}+3az^{-1}-8a^{-1}z^{-1}+7a^{-3}z^{-1}-2a^{-5}z^{-1}+az^{-3}-3a^{-1}z^{-3}+3a^{-3}z^{-3}-a^{-5}z^{-3}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-6z^{8}a^{-2}-2z^{8}a^{-4}-4z^{8}-5az^{7}-7z^{7}a^{-1}-3z^{7}a^{-3}-z^{7}a^{-5}-2a^{2}z^{6}+19z^{6}a^{-2}+7z^{6}a^{-4}+10z^{6}+17az^{5}+39z^{5}a^{-1}+27z^{5}a^{-3}+5z^{5}a^{-5}+2a^{2}z^{4}-6z^{4}a^{-2}-3z^{4}a^{-4}-z^{4}-4a^{3}z^{3}-24az^{3}-52z^{3}a^{-1}-42z^{3}a^{-3}-10z^{3}a^{-5}-a^{4}z^{2}-2a^{2}z^{2}-18z^{2}a^{-2}-9z^{2}a^{-4}-10z^{2}+2a^{3}z+16az+31za^{-1}+27za^{-3}+10za^{-5}+19a^{-2}+10a^{-4}+10-5az^{-1}-12a^{-1}z^{-1}-12a^{-3}z^{-1}-5a^{-5}z^{-1}-6a^{-2}z^{-2}-3a^{-4}z^{-2}-3z^{-2}+az^{-3}+3a^{-1}z^{-3}+3a^{-3}z^{-3}+a^{-5}z^{-3}$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

χ

12

1

10

1

-1

8

5

1

4

6

2

3

1

4

7

3

4

2

1

3

4

2

0

6

5

1

-2

1

2

4

3

-4

3

4

-1

-6

3

-8

1

-1

Integral Khovanov Homology

(db, data source)

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