L10a63

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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_16,11,17,12 X_18,13,19,14 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, -7, 9, -8, 5, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{4}-2u^{2}v^{3}+2u^{2}v^{2}-2u^{2}v+u^{2}-2uv^{4}+5uv^{3}-5uv^{2}+5uv-2u+v^{4}-2v^{3}+2v^{2}-2v+1}{uv^{2}}}$ (db)
Jones polynomial	${\frac {11}{q^{9/2}}}-{\frac {12}{q^{7/2}}}+{\frac {10}{q^{5/2}}}+q^{3/2}-{\frac {9}{q^{3/2}}}+{\frac {1}{q^{17/2}}}-{\frac {3}{q^{15/2}}}+{\frac {6}{q^{13/2}}}-{\frac {9}{q^{11/2}}}-3{\sqrt {q}}+{\frac {5}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{7}-2za^{7}-a^{7}z^{-1}+2z^{5}a^{5}+7z^{3}a^{5}+8za^{5}+3a^{5}z^{-1}-z^{7}a^{3}-5z^{5}a^{3}-10z^{3}a^{3}-9za^{3}-2a^{3}z^{-1}+z^{5}a+3z^{3}a+2za$ (db)
Kauffman polynomial	$-z^{4}a^{10}+z^{2}a^{10}-3z^{5}a^{9}+3z^{3}a^{9}-5z^{6}a^{8}+6z^{4}a^{8}-3z^{2}a^{8}+a^{8}-6z^{7}a^{7}+10z^{5}a^{7}-10z^{3}a^{7}+5za^{7}-a^{7}z^{-1}-4z^{8}a^{6}+2z^{6}a^{6}+6z^{4}a^{6}-9z^{2}a^{6}+3a^{6}-z^{9}a^{5}-10z^{7}a^{5}+32z^{5}a^{5}-35z^{3}a^{5}+16za^{5}-3a^{5}z^{-1}-7z^{8}a^{4}+14z^{6}a^{4}-3z^{4}a^{4}-6z^{2}a^{4}+3a^{4}-z^{9}a^{3}-7z^{7}a^{3}+29z^{5}a^{3}-32z^{3}a^{3}+15za^{3}-2a^{3}z^{-1}-3z^{8}a^{2}+6z^{6}a^{2}+z^{4}a^{2}-3z^{2}a^{2}-3z^{7}a+10z^{5}a-10z^{3}a+4za-z^{6}+3z^{4}-2z^{2}$ (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

4

1

-1

2

0

3

1

-2

6

2

4

-4

5

4

-1

-6

7

5

2

-8

5

6

1

-10

4

6

-2

-12

2

5

3

-14

1

4

-3

-16

2

-18

1

-1

Integral Khovanov Homology

(db, data source)

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