L10a170

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,6,13,5 X₈₄₉₃ X_2,16,3,15 X_16,7,17,8 X_14,19,11,20 X_20,13,15,14 X_18,10,19,9 X_10,12,5,11 X_4,17,1,18
Gauss code	{1, -4, 3, -10}, {9, -2, 7, -6}, {2, -1, 5, -3, 8, -9}, {4, -5, 10, -8, 6, -7}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(w-1)(x-1)(vwx+v(-w)+v-wx+x-1)}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	$-4q^{9/2}+{\frac {1}{q^{9/2}}}+8q^{7/2}-{\frac {4}{q^{7/2}}}-13q^{5/2}+{\frac {7}{q^{5/2}}}+14q^{3/2}-{\frac {13}{q^{3/2}}}+q^{11/2}-18{\sqrt {q}}+{\frac {13}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-3}-a^{3}z^{3}+2z^{3}a^{-3}+a^{3}z^{-3}-a^{-3}z^{-3}-a^{3}z+za^{-3}+a^{3}z^{-1}-a^{-3}z^{-1}-z^{7}a^{-1}+2az^{5}-4z^{5}a^{-1}+5az^{3}-6z^{3}a^{-1}-3az^{-3}+3a^{-1}z^{-3}+2az-2za^{-1}-3az^{-1}+3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}+4z^{5}a^{-5}-2z^{3}a^{-5}+a^{4}z^{6}+8z^{6}a^{-4}-2a^{4}z^{4}-7z^{4}a^{-4}+a^{4}z^{2}+z^{2}a^{-4}+4a^{3}z^{7}+11z^{7}a^{-3}-11a^{3}z^{5}-17z^{5}a^{-3}+10a^{3}z^{3}+12z^{3}a^{-3}+a^{3}z^{-3}+a^{-3}z^{-3}-3a^{3}z-3za^{-3}-2a^{3}z^{-1}-2a^{-3}z^{-1}+5a^{2}z^{8}+8z^{8}a^{-2}-9a^{2}z^{6}-6z^{6}a^{-2}-a^{2}z^{4}-6z^{4}a^{-2}+3a^{2}z^{2}+3z^{2}a^{-2}-3a^{2}z^{-2}-3a^{-2}z^{-2}+4a^{2}+4a^{-2}+2az^{9}+2z^{9}a^{-1}+11az^{7}+18z^{7}a^{-1}-43az^{5}-53z^{5}a^{-1}+40az^{3}+44z^{3}a^{-1}+3az^{-3}+3a^{-1}z^{-3}-11az-11za^{-1}-3az^{-1}-3a^{-1}z^{-1}+13z^{8}-24z^{6}+3z^{4}+4z^{2}-6z^{-2}+7$ (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

χ

12

1

-1

10

3

8

5

1

-4

6

8

3

5

4

8

7

-1

2

10

6

4

0

7

12

5

-2

6

0

-4

3

9

6

-6

1

4

-3

-8

3

-10

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$