L11a535

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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_16,7,17,8 X_8,15,5,16 X_22,18,15,17 X_20,14,21,13 X_12,20,13,19 X_14,22,9,21
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -10, 9, -11}, {7, -6, 8, -3, 10, -9, 11, -8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {-2t(1)t(4)^{2}t(3)^{2}+t(1)t(2)t(4)^{2}t(3)^{2}-2t(2)t(4)^{2}t(3)^{2}+2t(4)^{2}t(3)^{2}+t(1)t(4)t(3)^{2}-t(1)t(2)t(4)t(3)^{2}+2t(2)t(4)t(3)^{2}-t(4)t(3)^{2}+2t(1)t(4)^{2}t(3)-t(1)t(2)t(4)^{2}t(3)+t(2)t(4)^{2}t(3)-t(4)^{2}t(3)+t(1)t(3)-t(1)t(2)t(3)+2t(2)t(3)-3t(1)t(4)t(3)+2t(1)t(2)t(4)t(3)-3t(2)t(4)t(3)+2t(4)t(3)-t(3)-2t(1)+2t(1)t(2)-2t(2)+2t(1)t(4)-t(1)t(2)t(4)+t(2)t(4)-t(4)+1}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$-4q^{9/2}+{\frac {1}{q^{9/2}}}+7q^{7/2}-{\frac {5}{q^{7/2}}}-11q^{5/2}+{\frac {6}{q^{5/2}}}+13q^{3/2}-{\frac {12}{q^{3/2}}}+q^{11/2}-{\frac {1}{q^{11/2}}}-15{\sqrt {q}}+{\frac {12}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+12az^{3}-6z^{3}a^{-1}+2z^{3}a^{-3}+a^{5}z-11a^{3}z+16az-6za^{-1}+3a^{5}z^{-1}-10a^{3}z^{-1}+11az^{-1}-4a^{-1}z^{-1}+a^{5}z^{-3}-3a^{3}z^{-3}+3az^{-3}-a^{-1}z^{-3}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-a^{3}z^{9}-5az^{9}-4z^{9}a^{-1}-a^{4}z^{8}+a^{2}z^{8}-7z^{8}a^{-2}-5z^{8}-a^{5}z^{7}-a^{3}z^{7}+12az^{7}+4z^{7}a^{-1}-8z^{7}a^{-3}+2a^{4}z^{6}-2a^{2}z^{6}+10z^{6}a^{-2}-7z^{6}a^{-4}+13z^{6}+6a^{5}z^{5}+17a^{3}z^{5}-2az^{5}+9z^{5}a^{-3}-4z^{5}a^{-5}+6a^{4}z^{4}+19a^{2}z^{4}-4z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}+z^{4}-13a^{5}z^{3}-32a^{3}z^{3}-16az^{3}+3z^{3}a^{-5}-17a^{4}z^{2}-33a^{2}z^{2}-16z^{2}+13a^{5}z+28a^{3}z+21az+3za^{-1}-3za^{-3}+13a^{4}+24a^{2}-a^{-2}+11-6a^{5}z^{-1}-14a^{3}z^{-1}-12az^{-1}-3a^{-1}z^{-1}+a^{-3}z^{-1}-3a^{4}z^{-2}-6a^{2}z^{-2}-3z^{-2}+a^{5}z^{-3}+3a^{3}z^{-3}+3az^{-3}+a^{-1}z^{-3}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

3

8

4

1

-3

6

7

3

4

9

7

-2

2

6

4

2

0

7

10

3

-2

5

0

-4

2

8

6

-6

3

4

-1

-8

1

5

4

-10

0

-12

1

Integral Khovanov Homology

(db, data source)

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