L11n105

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,13,4,12 X_7,16,8,17 X_17,22,18,5 X_13,18,14,19 X_9,21,10,20 X_19,14,20,15 X_21,9,22,8 X_15,10,16,11 X₂₅₃₆ X_11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 8, -6, 9, -11, 2, -5, 7, -9, 3, -4, 5, -7, 6, -8, 4}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)^{3}\left(t(2)^{2}+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	${\frac {8}{q^{9/2}}}-{\frac {10}{q^{7/2}}}-q^{5/2}+{\frac {10}{q^{5/2}}}+3q^{3/2}-{\frac {11}{q^{3/2}}}+{\frac {2}{q^{13/2}}}-{\frac {5}{q^{11/2}}}-6{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+z^{5}a^{5}+4z^{3}a^{5}+6za^{5}+3a^{5}z^{-1}-z^{7}a^{3}-5z^{5}a^{3}-10z^{3}a^{3}-10za^{3}-3a^{3}z^{-1}+2z^{5}a+7z^{3}a+7za+2az^{-1}-z^{3}a^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$3a^{8}z^{2}-a^{8}+a^{7}z^{5}+5a^{7}z^{3}-3a^{7}z+a^{7}z^{-1}+5a^{6}z^{6}-7a^{6}z^{4}+6a^{6}z^{2}-2a^{6}+9a^{5}z^{7}-26a^{5}z^{5}+28a^{5}z^{3}-14a^{5}z+3a^{5}z^{-1}+7a^{4}z^{8}-18a^{4}z^{6}+11a^{4}z^{4}-3a^{4}z^{2}+2a^{3}z^{9}+7a^{3}z^{7}-42a^{3}z^{5}+48a^{3}z^{3}-22a^{3}z+3a^{3}z^{-1}+10a^{2}z^{8}-35a^{2}z^{6}+32a^{2}z^{4}-10a^{2}z^{2}+2a^{2}+2az^{9}-az^{7}+z^{7}a^{-1}-19az^{5}-4z^{5}a^{-1}+31az^{3}+6z^{3}a^{-1}-15az-4za^{-1}+2az^{-1}+a^{-1}z^{-1}+3z^{8}-12z^{6}+14z^{4}-4z^{2}$ (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

6

1

4

2

-2

2

4

1

3

0

4

2

-2

7

4

3

-4

5

6

1

-6

5

0

-8

3

5

2

-10

2

5

-3

-12

3

-14

2

-2

Integral Khovanov Homology

(db, data source)

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