L11n242

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X_16,8,17,7 X_10,5,1,6 X₆₃₇₄ X_4,9,5,10 X_13,18,14,19 X_19,22,20,11 X_15,21,16,20 X_21,15,22,14 X_2,11,3,12 X_8,18,9,17
Gauss code	{1, -10, 4, -5, 3, -4, 2, -11, 5, -3}, {10, -1, -6, 9, -8, -2, 11, 6, -7, 8, -9, 7}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)t(1)^{2}+t(2)^{2}t(1)-4t(2)t(1)+t(1)+t(2)\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {9}{q^{9/2}}}+{\frac {10}{q^{7/2}}}-{\frac {11}{q^{5/2}}}-2q^{3/2}+{\frac {10}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {6}{q^{11/2}}}+4{\sqrt {q}}-{\frac {8}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)-a^{7}z^{-1}+3a^{5}z^{3}+6a^{5}z+4a^{5}z^{-1}-2a^{3}z^{5}-7a^{3}z^{3}-11a^{3}z-6a^{3}z^{-1}+4az^{3}+8az+5az^{-1}-2za^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{5}z^{9}-a^{3}z^{9}-3a^{6}z^{8}-7a^{4}z^{8}-4a^{2}z^{8}-3a^{7}z^{7}-8a^{5}z^{7}-9a^{3}z^{7}-4az^{7}-a^{8}z^{6}+4a^{6}z^{6}+15a^{4}z^{6}+9a^{2}z^{6}-z^{6}+9a^{7}z^{5}+33a^{5}z^{5}+36a^{3}z^{5}+12az^{5}+3a^{8}z^{4}+8a^{6}z^{4}-3a^{4}z^{4}-10a^{2}z^{4}-2z^{4}-8a^{7}z^{3}-35a^{5}z^{3}-50a^{3}z^{3}-26az^{3}-3z^{3}a^{-1}-3a^{8}z^{2}-9a^{6}z^{2}-7a^{4}z^{2}+z^{2}+4a^{7}z+18a^{5}z+29a^{3}z+20az+5za^{-1}+a^{8}+3a^{6}+3a^{4}+a^{2}+1-a^{7}z^{-1}-4a^{5}z^{-1}-6a^{3}z^{-1}-5az^{-1}-2a^{-1}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

4

2

-2

0

6

2

4

-2

6

4

-2

-4

5

4

1

-6

5

6

1

-8

4

5

-1

-10

2

5

3

-12

1

4

-3

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

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