L10a56

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_16,7,17,8 X_10,4,11,3 X_12,5,13,6 X_20,12,7,11 X_18,13,19,14 X_2,15,3,16 X_4,19,5,20 X_14,10,15,9 X_6,18,1,17
Gauss code	{1, -7, 3, -8, 4, -10}, {2, -1, 9, -3, 5, -4, 6, -9, 7, -2, 10, -6, 8, -5}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{4}-4u^{2}v^{3}+6u^{2}v^{2}-4u^{2}v+u^{2}-2uv^{4}+8uv^{3}-13uv^{2}+8uv-2u+v^{4}-4v^{3}+6v^{2}-4v+1}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {11}{q^{9/2}}}-q^{7/2}+{\frac {16}{q^{7/2}}}+5q^{5/2}-{\frac {21}{q^{5/2}}}-11q^{3/2}+{\frac {22}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {5}{q^{11/2}}}+16{\sqrt {q}}-{\frac {21}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+3az^{5}-z^{5}a^{-1}+a^{5}z^{3}-3a^{3}z^{3}+3az^{3}-z^{3}a^{-1}-az+a^{3}z^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-5a^{3}z^{9}-5az^{9}-12a^{4}z^{8}-24a^{2}z^{8}-12z^{8}-11a^{5}z^{7}-14a^{3}z^{7}-14az^{7}-11z^{7}a^{-1}-5a^{6}z^{6}+16a^{4}z^{6}+42a^{2}z^{6}-5z^{6}a^{-2}+16z^{6}-a^{7}z^{5}+16a^{5}z^{5}+39a^{3}z^{5}+39az^{5}+16z^{5}a^{-1}-z^{5}a^{-3}+4a^{6}z^{4}-4a^{4}z^{4}-16a^{2}z^{4}+4z^{4}a^{-2}-4z^{4}-6a^{5}z^{3}-18a^{3}z^{3}-18az^{3}-6z^{3}a^{-1}-a^{3}z-az-a^{2}+a^{3}z^{-1}+az^{-1}$ (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

4

-4

4

7

1

6

2

9

4

-5

0

12

7

5

-2

11

10

-1

-4

10

11

-1

-6

7

12

5

-8

4

9

-5

-10

1

7

6

-12

4

-4

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-3$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{12}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$