L11a492

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_22,16,19,15 X_20,9,21,10 X_8,19,9,20 X_4,17,1,18 X_12,6,13,5 X_10,4,11,3 X_18,12,5,11 X_16,22,17,21 X_2,14,3,13
Gauss code	{1, -11, 8, -6}, {5, -4, 10, -3}, {7, -1, 2, -5, 4, -8, 9, -7, 11, -2, 3, -10, 6, -9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w} , ...)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{(u-1) (v-1) (w-1) \left(w^2-w+1\right)^2}{\sqrt{u} \sqrt{v} w^{5/2}}} (db)
Jones polynomial	$q^{6}-4q^{5}+9q^{4}-15q^{3}+20q^{2}-22q+25-19q^{-1}+15q^{-2}-9q^{-3}+4q^{-4}-q^{-5}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^4 a^{-4} +2 z^2 a^{-4} + a^{-4} -a^2 z^6-2 z^6 a^{-2} -3 a^2 z^4-7 z^4 a^{-2} -3 a^2 z^2-8 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -a^2-3 a^{-2} +z^8+5 z^6+10 z^4+9 z^2-2 z^{-2} +3} (db)
Kauffman polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 z^{10} a^{-2} +2 z^{10}+8 a z^9+14 z^9 a^{-1} +6 z^9 a^{-3} +11 a^2 z^8+16 z^8 a^{-2} +7 z^8 a^{-4} +20 z^8+8 a^3 z^7-8 a z^7-24 z^7 a^{-1} -4 z^7 a^{-3} +4 z^7 a^{-5} +4 a^4 z^6-23 a^2 z^6-55 z^6 a^{-2} -15 z^6 a^{-4} +z^6 a^{-6} -66 z^6+a^5 z^5-12 a^3 z^5-5 a z^5-17 z^5 a^{-3} -9 z^5 a^{-5} -5 a^4 z^4+24 a^2 z^4+60 z^4 a^{-2} +9 z^4 a^{-4} -2 z^4 a^{-6} +78 z^4-a^5 z^3+4 a^3 z^3+13 a z^3+21 z^3 a^{-1} +19 z^3 a^{-3} +6 z^3 a^{-5} -13 a^2 z^2-30 z^2 a^{-2} -6 z^2 a^{-4} +z^2 a^{-6} -36 z^2-a^3 z-5 a z-9 z a^{-1} -7 z a^{-3} -2 z a^{-5} +2 a^2+6 a^{-2} +2 a^{-4} +7-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2} } (db)

Khovanov Homology

The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ).

\		r
	\
j		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

13

1

11

3

-3

9

6

1

5

7

9

3

-6

5

11

6

5

3

11

9

-2

1

14

11

3

-1

9

15

6

-3

6

10

-4

-5

3

9

6

-7

1

6

-5

-9

3

-11

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{14}$
$r=1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=6}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}