L11n402

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_22,16,19,15 X_20,8,21,7 X_8,20,9,19 X_13,18,14,5 X_11,14,12,15 X_17,12,18,13 X_16,22,17,21 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {5, -4, 9, -3}, {10, -1, 4, -5, 11, -2, -7, 8, -6, 7, 3, -9, -8, 6}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(w-1)\left(uvw^{3}-uvw^{2}+uvw-uv-2uw^{3}+uw^{2}-uw-vw^{3}+vw^{2}-2vw-w^{4}+w^{3}-w^{2}+w\right)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$-2q^{-6}+5q^{-5}-8q^{-4}+q^{3}+11q^{-3}-2q^{2}-10q^{-2}+6q+11q^{-1}-8$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{6}z^{-2}-a^{6}-z^{4}a^{4}+z^{2}a^{4}+4a^{4}z^{-2}+6a^{4}+z^{6}a^{2}+2z^{4}a^{2}-2z^{2}a^{2}-5a^{2}z^{-2}-7a^{2}-2z^{4}-4z^{2}+2z^{-2}+z^{2}a^{-2}+2a^{-2}$ (db)
Kauffman polynomial	$a^{3}z^{9}+az^{9}+4a^{4}z^{8}+7a^{2}z^{8}+3z^{8}+4a^{5}z^{7}+8a^{3}z^{7}+6az^{7}+2z^{7}a^{-1}+a^{6}z^{6}-9a^{4}z^{6}-19a^{2}z^{6}+z^{6}a^{-2}-8z^{6}-9a^{5}z^{5}-29a^{3}z^{5}-25az^{5}-5z^{5}a^{-1}+4a^{6}z^{4}+19a^{4}z^{4}+27a^{2}z^{4}-4z^{4}a^{-2}+8z^{4}+3a^{7}z^{3}+20a^{5}z^{3}+43a^{3}z^{3}+28az^{3}+2z^{3}a^{-1}-4a^{6}z^{2}-24a^{4}z^{2}-34a^{2}z^{2}+5z^{2}a^{-2}-9z^{2}-3a^{7}z-17a^{5}z-33a^{3}z-18az+za^{-1}+3a^{6}+16a^{4}+21a^{2}-2a^{-2}+7+a^{7}z^{-1}+5a^{5}z^{-1}+9a^{3}z^{-1}+5az^{-1}-a^{6}z^{-2}-4a^{4}z^{-2}-5a^{2}z^{-2}-2z^{-2}$ (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

χ

7

1

5

2

1

-1

3

4

1

4

2

-2

-1

7

4

3

-3

5

6

1

-5

6

5

1

-7

2

5

3

-9

3

6

-3

-11

3

-13

2

-2

Integral Khovanov Homology

(db, data source)

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