L11n3

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Link L11n3.

A graph, L11n3.

A part of a knot and a part of a graph.

Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_5,10,6,11 X₈₄₉₃ X_11,20,12,21 X_17,5,18,22 X_21,19,22,18 X_19,12,20,13 X_9,16,10,17 X_2,14,3,13
Gauss code	{1, -11, 5, -3}, {-4, -1, 2, -5, -10, 4, -6, 9, 11, -2, 3, 10, -7, 8, -9, 6, -8, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(v^{2}-3v+1\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {6}{q^{9/2}}}+{\frac {6}{q^{7/2}}}-{\frac {7}{q^{5/2}}}-q^{3/2}+{\frac {6}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {2}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+2{\sqrt {q}}-{\frac {5}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)-a^{7}z^{-1}+2a^{5}z^{3}+4a^{5}z+3a^{5}z^{-1}-a^{3}z^{5}-3a^{3}z^{3}-5a^{3}z-3a^{3}z^{-1}+2az^{3}+3az+2az^{-1}-za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{5}z^{9}-a^{3}z^{9}-2a^{6}z^{8}-4a^{4}z^{8}-2a^{2}z^{8}-2a^{7}z^{7}+a^{3}z^{7}-az^{7}-a^{8}z^{6}+5a^{6}z^{6}+14a^{4}z^{6}+8a^{2}z^{6}+7a^{7}z^{5}+9a^{5}z^{5}+5a^{3}z^{5}+3az^{5}+4a^{8}z^{4}+a^{6}z^{4}-16a^{4}z^{4}-15a^{2}z^{4}-2z^{4}-6a^{7}z^{3}-13a^{5}z^{3}-15a^{3}z^{3}-9az^{3}-z^{3}a^{-1}-4a^{8}z^{2}-4a^{6}z^{2}+6a^{4}z^{2}+8a^{2}z^{2}+2z^{2}+3a^{7}z+10a^{5}z+12a^{3}z+7az+2za^{-1}+a^{8}+2a^{6}-2a^{2}-a^{7}z^{-1}-3a^{5}z^{-1}-3a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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

1

2

1

-1

0

4

1

3

-2

4

3

-1

-4

3

2

1

-6

3

4

1

-8

3

0

-10

1

3

2

-12

1

3

-2

-14

1

-16

1

-1

Integral Khovanov Homology

(db, data source)

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