L10a49

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_10,5,11,6 X_14,3,15,4 X_18,11,19,12 X_20,13,5,14 X_12,19,13,20 X_2,9,3,10 X_8,15,9,16
Gauss code	{1, -9, 5, -3}, {4, -1, 2, -10, 9, -4, 6, -8, 7, -5, 10, -2, 3, -6, 8, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(2)^{5}+4t(1)t(2)^{4}-4t(2)^{4}-6t(1)t(2)^{3}+6t(2)^{3}+6t(1)t(2)^{2}-6t(2)^{2}-4t(1)t(2)+4t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {7}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{5/2}}}-{\frac {1}{q^{25/2}}}+{\frac {4}{q^{23/2}}}-{\frac {7}{q^{21/2}}}+{\frac {11}{q^{19/2}}}-{\frac {13}{q^{17/2}}}+{\frac {14}{q^{15/2}}}-{\frac {14}{q^{13/2}}}+{\frac {9}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z^{3}a^{11}-2a^{11}z^{-1}-z^{5}a^{9}+z^{3}a^{9}+8za^{9}+5a^{9}z^{-1}-3z^{5}a^{7}-10z^{3}a^{7}-10za^{7}-3a^{7}z^{-1}-z^{5}a^{5}-2z^{3}a^{5}$ (db)
Kauffman polynomial	$-z^{5}a^{15}+z^{3}a^{15}-4z^{6}a^{14}+7z^{4}a^{14}-2z^{2}a^{14}-a^{14}-6z^{7}a^{13}+10z^{5}a^{13}-3z^{3}a^{13}+za^{13}-5z^{8}a^{12}+5z^{6}a^{12}+z^{4}a^{12}+z^{2}a^{12}-2z^{9}a^{11}-6z^{7}a^{11}+16z^{5}a^{11}-11z^{3}a^{11}+5za^{11}-2a^{11}z^{-1}-10z^{8}a^{10}+20z^{6}a^{10}-13z^{4}a^{10}-4z^{2}a^{10}+5a^{10}-2z^{9}a^{9}-6z^{7}a^{9}+22z^{5}a^{9}-27z^{3}a^{9}+14za^{9}-5a^{9}z^{-1}-5z^{8}a^{8}+8z^{6}a^{8}-2z^{4}a^{8}-7z^{2}a^{8}+5a^{8}-6z^{7}a^{7}+16z^{5}a^{7}-18z^{3}a^{7}+10za^{7}-3a^{7}z^{-1}-3z^{6}a^{6}+5z^{4}a^{6}-z^{5}a^{5}+2z^{3}a^{5}$ (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

4

-10

5

3

-2

-12

9

4

5

-14

6

0

-16

7

8

-1

-18

4

6

2

-20

3

7

-4

-22

1

4

3

-24

3

-3

-26

1

Integral Khovanov Homology

(db, data source)

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