L10a38

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {\left(v^{2}-v+1\right)\left(3uv^{2}-3uv+u+v^{3}-3v^{2}+3v\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{5/2}}}-{\frac {1}{q^{25/2}}}+{\frac {3}{q^{23/2}}}-{\frac {6}{q^{21/2}}}+{\frac {11}{q^{19/2}}}-{\frac {13}{q^{17/2}}}+{\frac {14}{q^{15/2}}}-{\frac {14}{q^{13/2}}}+{\frac {10}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{13}z^{-1}-4za^{11}-5a^{11}z^{-1}+6z^{3}a^{9}+14za^{9}+8a^{9}z^{-1}-3z^{5}a^{7}-10z^{3}a^{7}-11za^{7}-4a^{7}z^{-1}-z^{5}a^{5}-2z^{3}a^{5}-za^{5}$ (db)
Kauffman polynomial	$a^{15}z^{5}-2a^{15}z^{3}+a^{15}z+3a^{14}z^{6}-6a^{14}z^{4}+5a^{14}z^{2}-2a^{14}+4a^{13}z^{7}-4a^{13}z^{5}-a^{13}z^{3}+a^{13}z+a^{13}z^{-1}+3a^{12}z^{8}+5a^{12}z^{6}-21a^{12}z^{4}+22a^{12}z^{2}-9a^{12}+a^{11}z^{9}+11a^{11}z^{7}-23a^{11}z^{5}+16a^{11}z^{3}-9a^{11}z+5a^{11}z^{-1}+7a^{10}z^{8}+a^{10}z^{6}-29a^{10}z^{4}+35a^{10}z^{2}-14a^{10}+a^{9}z^{9}+13a^{9}z^{7}-32a^{9}z^{5}+32a^{9}z^{3}-22a^{9}z+8a^{9}z^{-1}+4a^{8}z^{8}+2a^{8}z^{6}-18a^{8}z^{4}+19a^{8}z^{2}-8a^{8}+6a^{7}z^{7}-13a^{7}z^{5}+15a^{7}z^{3}-12a^{7}z+4a^{7}z^{-1}+3a^{6}z^{6}-4a^{6}z^{4}+a^{6}z^{2}+a^{5}z^{5}-2a^{5}z^{3}+a^{5}z$ (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		\

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

3

1

-2

-8

5

-10

5

3

-2

-12

9

5

4

-14

6

0

-16

7

8

-1

-18

4

6

2

-20

2

7

-5

-22

1

4

3

-24

2

-2

-26

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-6$	$i=-4$
$r=-10$	${\mathbb {Z} }$
$r=-9$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-7$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-6$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-5$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$