L10a32

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Link Presentations

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

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

Polynomial invariants

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

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

2

-2

10

4

1

3

8

5

2

-3

6

7

4

3

4

5

0

2

6

7

-1

0

5

7

2

-2

1

4

-3

-4

1

5

4

-6

1

-1

-8

1

Integral Khovanov Homology

(db, data source)

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