L10a7

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_4,19,1,20 X_14,12,15,11 X₈₄₉₃ X_12,5,13,6 X_20,13,5,14 X_16,10,17,9 X_10,16,11,15 X_2,18,3,17
Gauss code	{1, -10, 5, -3}, {6, -1, 2, -5, 8, -9, 4, -6, 7, -4, 9, -8, 10, -2, 3, -7}

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	$q^{9/2}-{\frac {3}{q^{9/2}}}-4q^{7/2}+{\frac {5}{q^{7/2}}}+6q^{5/2}-{\frac {9}{q^{5/2}}}-9q^{3/2}+{\frac {11}{q^{3/2}}}+{\frac {1}{q^{11/2}}}+11{\sqrt {q}}-{\frac {12}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}(-z)+2a^{3}z^{3}+z^{3}a^{-3}+2a^{3}z+a^{3}z^{-1}-a^{-3}z^{-1}-az^{5}-z^{5}a^{-1}-az^{3}-z^{3}a^{-1}-2az-2az^{-1}+za^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2az^{9}-2z^{9}a^{-1}-4a^{2}z^{8}-5z^{8}a^{-2}-9z^{8}-4a^{3}z^{7}-2az^{7}-2z^{7}a^{-1}-4z^{7}a^{-3}-4a^{4}z^{6}+4a^{2}z^{6}+15z^{6}a^{-2}-z^{6}a^{-4}+24z^{6}-3a^{5}z^{5}+8az^{5}+17z^{5}a^{-1}+12z^{5}a^{-3}-a^{6}z^{4}+3a^{4}z^{4}-2a^{2}z^{4}-11z^{4}a^{-2}+2z^{4}a^{-4}-19z^{4}+4a^{5}z^{3}+7a^{3}z^{3}-az^{3}-11z^{3}a^{-1}-7z^{3}a^{-3}+a^{6}z^{2}+a^{2}z^{2}+2z^{2}a^{-2}+4z^{2}-2a^{5}z-5a^{3}z-6az-4za^{-1}-za^{-3}+1+a^{3}z^{-1}+2az^{-1}+2a^{-1}z^{-1}+a^{-3}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

3

6

3

1

-2

4

6

3

2

5

3

-2

0

7

6

1

-2

6

7

1

-4

3

5

-2

-6

2

6

4

-8

1

3

-2

-10

2

-12

1

-1

Integral Khovanov Homology

(db, data source)

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