L10a14

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)^{3}\left(t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+12{\sqrt {q}}-{\frac {16}{\sqrt {q}}}+{\frac {16}{q^{3/2}}}-{\frac {15}{q^{5/2}}}+{\frac {12}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+4az^{5}-z^{5}a^{-1}+a^{5}z^{3}-6a^{3}z^{3}+7az^{3}-2z^{3}a^{-1}+2a^{5}z-7a^{3}z+7az-2za^{-1}+a^{5}z^{-1}-3a^{3}z^{-1}+4az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{3}z^{9}-2az^{9}-5a^{4}z^{8}-12a^{2}z^{8}-7z^{8}-5a^{5}z^{7}-11a^{3}z^{7}-14az^{7}-8z^{7}a^{-1}-3a^{6}z^{6}+3a^{4}z^{6}+18a^{2}z^{6}-4z^{6}a^{-2}+8z^{6}-a^{7}z^{5}+7a^{5}z^{5}+30a^{3}z^{5}+38az^{5}+15z^{5}a^{-1}-z^{5}a^{-3}+5a^{6}z^{4}+5a^{4}z^{4}-4a^{2}z^{4}+5z^{4}a^{-2}+z^{4}+2a^{7}z^{3}-4a^{5}z^{3}-28a^{3}z^{3}-33az^{3}-10z^{3}a^{-1}+z^{3}a^{-3}-3a^{6}z^{2}-7a^{4}z^{2}-5a^{2}z^{2}-z^{2}a^{-2}-2z^{2}-a^{7}z+2a^{5}z+13a^{3}z+15az+5za^{-1}+a^{6}+3a^{4}+3a^{2}+2-a^{5}z^{-1}-3a^{3}z^{-1}-4az^{-1}-2a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

3

-3

4

6

1

5

2

6

3

-3

0

10

6

4

-2

8

0

-4

7

8

-1

-6

5

8

3

-8

2

7

-5

-10

1

5

4

-12

2

-2

-14

1

Integral Khovanov Homology

(db, data source)

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