L10a39

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(v^{2}-v+1\right)^{2}}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-3q^{9/2}+{\frac {1}{q^{9/2}}}+7q^{7/2}-{\frac {2}{q^{7/2}}}-10q^{5/2}+{\frac {5}{q^{5/2}}}+11q^{3/2}-{\frac {9}{q^{3/2}}}+q^{11/2}-13{\sqrt {q}}+{\frac {10}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-3}-a^{3}z^{3}+3z^{3}a^{-3}-3a^{3}z+4za^{-3}-2a^{3}z^{-1}+2a^{-3}z^{-1}-z^{7}a^{-1}+2az^{5}-5z^{5}a^{-1}+8az^{3}-11z^{3}a^{-1}+12az-13za^{-1}+7az^{-1}-7a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-2a^{2}z^{8}-4z^{8}a^{-2}-6z^{8}-2a^{3}z^{7}-5az^{7}-10z^{7}a^{-1}-7z^{7}a^{-3}-a^{4}z^{6}+a^{2}z^{6}-2z^{6}a^{-2}-6z^{6}a^{-4}+6z^{6}+6a^{3}z^{5}+19az^{5}+27z^{5}a^{-1}+11z^{5}a^{-3}-3z^{5}a^{-5}+4a^{4}z^{4}+11a^{2}z^{4}+17z^{4}a^{-2}+8z^{4}a^{-4}-z^{4}a^{-6}+15z^{4}-6a^{3}z^{3}-21az^{3}-27z^{3}a^{-1}-10z^{3}a^{-3}+2z^{3}a^{-5}-5a^{4}z^{2}-17a^{2}z^{2}-20z^{2}a^{-2}-6z^{2}a^{-4}+z^{2}a^{-6}-25z^{2}+4a^{3}z+16az+18za^{-1}+6za^{-3}+2a^{4}+8a^{2}+8a^{-2}+2a^{-4}+13-2a^{3}z^{-1}-7az^{-1}-7a^{-1}z^{-1}-2a^{-3}z^{-1}$ (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

12

1

-1

10

2

8

5

1

-4

6

5

2

3

4

6

5

-1

2

7

5

2

0

5

8

3

-2

4

5

-1

-4

1

5

4

-6

1

4

-3

-8

1

-10

1

-1

Integral Khovanov Homology

(db, data source)

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