L11n225

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_3,12,4,13 X_16,9,17,10 X_11,20,12,21 X_22,15,9,16 X_5,14,6,15 X_13,4,14,5 X_6,20,7,19 X_18,8,19,7 X_8,18,1,17 X_21,3,22,2
Gauss code	{1, 11, -2, 7, -6, -8, 9, -10}, {3, -1, -4, 2, -7, 6, 5, -3, 10, -9, 8, 4, -11, -5}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)t(1)^{2}+2t(2)^{2}t(1)-t(2)t(1)+2t(1)+t(2)\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-2{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {6}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {10}{q^{7/2}}}+{\frac {9}{q^{9/2}}}-{\frac {8}{q^{11/2}}}+{\frac {6}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}\left(-z^{3}\right)-a^{7}z+a^{5}z^{5}+a^{5}z^{3}-2a^{5}z-a^{5}z^{-1}+2a^{3}z^{5}+7a^{3}z^{3}+8a^{3}z+3a^{3}z^{-1}-2az^{3}-5az-2az^{-1}$ (db)
Kauffman polynomial	$-z^{4}a^{10}+z^{2}a^{10}-3z^{5}a^{9}+3z^{3}a^{9}-5z^{6}a^{8}+7z^{4}a^{8}-3z^{2}a^{8}-5z^{7}a^{7}+8z^{5}a^{7}-7z^{3}a^{7}+2za^{7}-3z^{8}a^{6}+3z^{6}a^{6}-4z^{4}a^{6}+4z^{2}a^{6}-a^{6}-z^{9}a^{5}-2z^{7}a^{5}+2z^{5}a^{5}+4z^{3}a^{5}-4za^{5}+a^{5}z^{-1}-4z^{8}a^{4}+10z^{6}a^{4}-14z^{4}a^{4}+12z^{2}a^{4}-3a^{4}-z^{9}a^{3}+3z^{7}a^{3}-12z^{5}a^{3}+24z^{3}a^{3}-15za^{3}+3a^{3}z^{-1}-z^{8}a^{2}+2z^{6}a^{2}-2z^{4}a^{2}+4z^{2}a^{2}-3a^{2}-3z^{5}a+10z^{3}a-9za+2az^{-1}$ (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

2

0

1

-1

-2

5

2

3

-4

5

3

-2

-6

5

3

2

-8

4

5

1

-10

4

5

-1

-12

2

4

2

-14

1

4

-3

-16

2

-18

1

-1

Integral Khovanov Homology

(db, data source)

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