L11a137

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_2,11,3,12 X_14,3,15,4 X_12,5,13,6 X_22,13,5,14 X_4,21,1,22 X_20,16,21,15 X_16,8,17,7 X_18,10,19,9 X_8,18,9,17 X_10,20,11,19
Gauss code	{1, -2, 3, -6}, {4, -1, 8, -10, 9, -11, 2, -4, 5, -3, 7, -8, 10, -9, 11, -7, 6, -5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(1)t(2)^{5}-2t(2)^{5}-5t(1)t(2)^{4}+6t(2)^{4}+6t(1)t(2)^{3}-6t(2)^{3}-6t(1)t(2)^{2}+6t(2)^{2}+6t(1)t(2)-5t(2)-2t(1)+2}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-4q^{9/2}+{\frac {3}{q^{9/2}}}+8q^{7/2}-{\frac {6}{q^{7/2}}}-12q^{5/2}+{\frac {9}{q^{5/2}}}+16q^{3/2}-{\frac {15}{q^{3/2}}}+q^{11/2}-{\frac {1}{q^{11/2}}}-17{\sqrt {q}}+{\frac {16}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-az^{7}-z^{7}a^{-1}+a^{3}z^{5}-4az^{5}-3z^{5}a^{-1}+z^{5}a^{-3}+3a^{3}z^{3}-6az^{3}-z^{3}a^{-1}+2z^{3}a^{-3}+2a^{3}z-6az+2za^{-1}+2a^{3}z^{-1}-3az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-4a^{3}z^{9}-10az^{9}-6z^{9}a^{-1}-3a^{4}z^{8}-9z^{8}a^{-2}-6z^{8}-a^{5}z^{7}+15a^{3}z^{7}+33az^{7}+7z^{7}a^{-1}-10z^{7}a^{-3}+12a^{4}z^{6}+17a^{2}z^{6}+11z^{6}a^{-2}-8z^{6}a^{-4}+24z^{6}+4a^{5}z^{5}-18a^{3}z^{5}-39az^{5}-z^{5}a^{-1}+12z^{5}a^{-3}-4z^{5}a^{-5}-13a^{4}z^{4}-23a^{2}z^{4}+8z^{4}a^{-4}-z^{4}a^{-6}-19z^{4}-4a^{5}z^{3}+11a^{3}z^{3}+24az^{3}+4z^{3}a^{-1}-3z^{3}a^{-3}+2z^{3}a^{-5}+3a^{4}z^{2}+10a^{2}z^{2}-2z^{2}a^{-2}-2z^{2}a^{-4}+7z^{2}+a^{5}z-5a^{3}z-8az-2za^{-1}-3a^{2}-a^{-2}-3+2a^{3}z^{-1}+3az^{-1}+a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

3

8

5

1

-4

6

7

3

4

9

5

-4

2

8

7

1

0

9

10

1

-2

6

7

-1

-4

3

9

6

-6

3

6

-3

-8

1

4

3

-10

2

-2

-12

1

Integral Khovanov Homology

(db, data source)

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