L11n1

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Link L11n1.

A graph, L11n1.

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 X₃₈₄₉ 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_13,2,14,3
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 $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{5}-4t(1)t(2)^{4}+2t(1)t(2)^{3}+2t(2)^{2}-4t(2)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {3}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {3}{q^{15/2}}}-{\frac {2}{q^{17/2}}}+{\frac {1}{q^{19/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+a^{9}z^{3}+5a^{9}z+3a^{9}z^{-1}-2a^{7}z^{5}-9a^{7}z^{3}-9a^{7}z-3a^{7}z^{-1}+a^{5}z^{7}+5a^{5}z^{5}+7a^{5}z^{3}+5a^{5}z+2a^{5}z^{-1}-a^{3}z^{5}-4a^{3}z^{3}-3a^{3}z-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{2}a^{12}+a^{12}-2z^{3}a^{11}+2za^{11}-a^{11}z^{-1}-z^{6}a^{10}+3z^{4}a^{10}-5z^{2}a^{10}+2a^{10}-2z^{7}a^{9}+9z^{5}a^{9}-16z^{3}a^{9}+12za^{9}-3a^{9}z^{-1}-2z^{8}a^{8}+8z^{6}a^{8}-9z^{4}a^{8}+4z^{2}a^{8}-z^{9}a^{7}+z^{7}a^{7}+12z^{5}a^{7}-25z^{3}a^{7}+17za^{7}-3a^{7}z^{-1}-4z^{8}a^{6}+19z^{6}a^{6}-25z^{4}a^{6}+12z^{2}a^{6}-2a^{6}-z^{9}a^{5}+2z^{7}a^{5}+8z^{5}a^{5}-18z^{3}a^{5}+11za^{5}-2a^{5}z^{-1}-2z^{8}a^{4}+10z^{6}a^{4}-13z^{4}a^{4}+4z^{2}a^{4}-z^{7}a^{3}+5z^{5}a^{3}-7z^{3}a^{3}+4za^{3}-a^{3}z^{-1}$ (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

0

1

-2

1

-1

-4

2

1

-6

1

3

2

0

-8

3

1

2

-10

1

3

1

-12

1

4

3

0

-14

1

2

1

-16

2

3

-1

-18

1

-20

1

-1

Integral Khovanov Homology

(db, data source)

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