L11a251

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_20,15,21,16 X_14,6,15,5 X_12,4,13,3 X_4,14,5,13 X_2,19,3,20 X_16,7,17,8 X_8,9,1,10 X_18,12,19,11 X_22,18,9,17 X_6,22,7,21
Gauss code	{1, -6, 4, -5, 3, -11, 7, -8}, {8, -1, 9, -4, 5, -3, 2, -7, 10, -9, 6, -2, 11, -10}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)^{3}(uv+1)^{2}}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-q^{13/2}+4q^{11/2}-8q^{9/2}+13q^{7/2}-18q^{5/2}+20q^{3/2}-21{\sqrt {q}}+{\frac {17}{\sqrt {q}}}-{\frac {13}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-3}-4z^{5}a^{-3}-4z^{3}a^{-3}+z^{9}a^{-1}-az^{7}+6z^{7}a^{-1}-4az^{5}+12z^{5}a^{-1}-4az^{3}+8z^{3}a^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-3z^{10}a^{-2}-3z^{10}-7az^{9}-14z^{9}a^{-1}-7z^{9}a^{-3}-7a^{2}z^{8}-4z^{8}a^{-2}-8z^{8}a^{-4}-3z^{8}-4a^{3}z^{7}+18az^{7}+41z^{7}a^{-1}+12z^{7}a^{-3}-7z^{7}a^{-5}-a^{4}z^{6}+19a^{2}z^{6}+20z^{6}a^{-2}+12z^{6}a^{-4}-4z^{6}a^{-6}+24z^{6}+10a^{3}z^{5}-15az^{5}-48z^{5}a^{-1}-11z^{5}a^{-3}+11z^{5}a^{-5}-z^{5}a^{-7}+2a^{4}z^{4}-13a^{2}z^{4}-23z^{4}a^{-2}-4z^{4}a^{-4}+6z^{4}a^{-6}-28z^{4}-4a^{3}z^{3}+6az^{3}+19z^{3}a^{-1}+5z^{3}a^{-3}-3z^{3}a^{-5}+z^{3}a^{-7}+3a^{2}z^{2}+6z^{2}a^{-2}-z^{2}a^{-6}+8z^{2}+1-az^{-1}-a^{-1}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

3

-3

10

5

1

4

8

3

-5

6

10

5

4

10

8

-2

2

11

10

1

0

8

12

4

-2

5

9

-4

3

8

5

-6

1

5

-4

-8

3

-10

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$