L10a165

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Link Presentations

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uvwx-2uvw-2uvx+2uv-2uwx+2uw+3ux-2u-2vwx+3vw+2vx-2v+2wx-2w-2x+1}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}{\sqrt {x}}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {11}{q^{7/2}}}+{\frac {10}{q^{9/2}}}-{\frac {11}{q^{11/2}}}+{\frac {5}{q^{13/2}}}-{\frac {5}{q^{15/2}}}+{\frac {1}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{11}z^{-3}-4a^{9}z^{-1}-3a^{9}z^{-3}+6za^{7}+8a^{7}z^{-1}+3a^{7}z^{-3}-4z^{3}a^{5}-6za^{5}-4a^{5}z^{-1}-a^{5}z^{-3}+z^{5}a^{3}+z^{3}a^{3}-z^{3}a$ (db)
Kauffman polynomial	$-z^{5}a^{11}+4z^{3}a^{11}-6za^{11}+4a^{11}z^{-1}-a^{11}z^{-3}-z^{6}a^{10}+6z^{2}a^{10}+3a^{10}z^{-2}-8a^{10}-z^{7}a^{9}-3z^{5}a^{9}+12z^{3}a^{9}-14za^{9}+9a^{9}z^{-1}-3a^{9}z^{-3}-z^{8}a^{8}-2z^{6}a^{8}+12z^{2}a^{8}+6a^{8}z^{-2}-15a^{8}-z^{9}a^{7}-6z^{5}a^{7}+13z^{3}a^{7}-14za^{7}+9a^{7}z^{-1}-3a^{7}z^{-3}-5z^{8}a^{6}+9z^{6}a^{6}-9z^{4}a^{6}+6z^{2}a^{6}+3a^{6}z^{-2}-8a^{6}-z^{9}a^{5}-5z^{7}a^{5}+10z^{5}a^{5}-z^{3}a^{5}-6za^{5}+4a^{5}z^{-1}-a^{5}z^{-3}-4z^{8}a^{4}+6z^{6}a^{4}-2z^{4}a^{4}-6z^{7}a^{3}+13z^{5}a^{3}-5z^{3}a^{3}-4z^{6}a^{2}+7z^{4}a^{2}-z^{5}a+z^{3}a$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

3

-3

-2

4

1

3

-4

5

4

-1

-6

6

3

-8

4

5

1

-10

7

6

1

-12

4

10

6

-14

1

0

-16

4

-18

1

0

-20

1

Integral Khovanov Homology

(db, data source)

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