L10a65

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_20,10,7,9 X₂₇₃₈ X_4,20,5,19 X_14,5,15,6 X_18,11,19,12 X_16,13,17,14 X_12,17,13,18 X_6,15,1,16
Gauss code	{1, -4, 2, -5, 6, -10}, {4, -1, 3, -2, 7, -9, 8, -6, 10, -8, 9, -7, 5, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(2)^{2}t(1)^{2}-4t(2)t(1)^{2}+2t(1)^{2}-4t(2)^{2}t(1)+9t(2)t(1)-4t(1)+2t(2)^{2}-4t(2)+2}{t(1)t(2)}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}+{\frac {10}{q^{7/2}}}+q^{5/2}-{\frac {11}{q^{5/2}}}-3q^{3/2}+{\frac {10}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {5}{q^{11/2}}}+5{\sqrt {q}}-{\frac {9}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}+2z^{3}a^{5}+2za^{5}-z^{5}a^{3}-z^{3}a^{3}+a^{3}z^{-1}-z^{5}a-2z^{3}a-3za-az^{-1}+z^{3}a^{-1}+za^{-1}$ (db)
Kauffman polynomial	$-a^{5}z^{9}-a^{3}z^{9}-3a^{6}z^{8}-6a^{4}z^{8}-3a^{2}z^{8}-3a^{7}z^{7}-5a^{5}z^{7}-6a^{3}z^{7}-4az^{7}-a^{8}z^{6}+7a^{6}z^{6}+12a^{4}z^{6}-4z^{6}+10a^{7}z^{5}+20a^{5}z^{5}+14a^{3}z^{5}+az^{5}-3z^{5}a^{-1}+3a^{8}z^{4}-2a^{6}z^{4}-6a^{4}z^{4}+3a^{2}z^{4}-z^{4}a^{-2}+3z^{4}-9a^{7}z^{3}-17a^{5}z^{3}-7a^{3}z^{3}+5az^{3}+4z^{3}a^{-1}-2a^{8}z^{2}-a^{6}z^{2}+2a^{4}z^{2}+z^{2}a^{-2}+2a^{7}z+4a^{5}z-a^{3}z-5az-2za^{-1}-a^{2}+a^{3}z^{-1}+az^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

6

1

-1

4

2

3

1

-2

0

6

2

4

-2

5

4

-1

-4

6

5

1

-6

5

6

1

-8

3

5

-2

-10

2

5

3

-12

1

3

-2

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

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