L10a18

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)^{3}\left(v^{2}+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	${\frac {10}{q^{9/2}}}-{\frac {11}{q^{7/2}}}+{\frac {9}{q^{5/2}}}+q^{3/2}-{\frac {9}{q^{3/2}}}+{\frac {1}{q^{17/2}}}-{\frac {2}{q^{15/2}}}+{\frac {5}{q^{13/2}}}-{\frac {8}{q^{11/2}}}-3{\sqrt {q}}+{\frac {5}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}\left(-z^{3}\right)-3a^{7}z-2a^{7}z^{-1}+2a^{5}z^{5}+8a^{5}z^{3}+11a^{5}z+5a^{5}z^{-1}-a^{3}z^{7}-5a^{3}z^{5}-10a^{3}z^{3}-10a^{3}z-3a^{3}z^{-1}+az^{5}+3az^{3}+2az$ (db)
Kauffman polynomial	$-z^{4}a^{10}+2z^{2}a^{10}-a^{10}-2z^{5}a^{9}+2z^{3}a^{9}-3z^{6}a^{8}+2z^{4}a^{8}-4z^{7}a^{7}+6z^{5}a^{7}-8z^{3}a^{7}+5za^{7}-2a^{7}z^{-1}-3z^{8}a^{6}+z^{6}a^{6}+6z^{4}a^{6}-11z^{2}a^{6}+5a^{6}-z^{9}a^{5}-7z^{7}a^{5}+24z^{5}a^{5}-29z^{3}a^{5}+17za^{5}-5a^{5}z^{-1}-6z^{8}a^{4}+11z^{6}a^{4}+2z^{4}a^{4}-11z^{2}a^{4}+5a^{4}-z^{9}a^{3}-6z^{7}a^{3}+26z^{5}a^{3}-29z^{3}a^{3}+16za^{3}-3a^{3}z^{-1}-3z^{8}a^{2}+6z^{6}a^{2}+2z^{4}a^{2}-4z^{2}a^{2}-3z^{7}a+10z^{5}a-10z^{3}a+4za-z^{6}+3z^{4}-2z^{2}$ (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

0

-6

6

4

2

-8

4

5

1

-10

4

6

-2

-12

1

4

3

-14

1

4

-3

-16

1

-18

1

-1

Integral Khovanov Homology

(db, data source)

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