L11a171

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_20,9,21,10 X_14,5,15,6 X_18,12,19,11 X_10,4,11,3 X_12,7,13,8 X_16,13,17,14 X_22,17,7,18 X_6,15,1,16 X_4,21,5,22 X_2,20,3,19
Gauss code	{1, -11, 5, -10, 3, -9}, {6, -1, 2, -5, 4, -6, 7, -3, 9, -7, 8, -4, 11, -2, 10, -8}

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{2 t(1)^2 t(2)^4-3 t(1) t(2)^4+t(2)^4-7 t(1)^2 t(2)^3+12 t(1) t(2)^3-5 t(2)^3+9 t(1)^2 t(2)^2-17 t(1) t(2)^2+9 t(2)^2-5 t(1)^2 t(2)+12 t(1) t(2)-7 t(2)+t(1)^2-3 t(1)+2}{t(1) t(2)^2}} (db)
Jones 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 q^{3/2}-5 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{20}{q^{3/2}}+\frac{26}{q^{5/2}}-\frac{31}{q^{7/2}}+\frac{31}{q^{9/2}}-\frac{27}{q^{11/2}}+\frac{20}{q^{13/2}}-\frac{12}{q^{15/2}}+\frac{5}{q^{17/2}}-\frac{1}{q^{19/2}}} (db)
Signature	-3 (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 a^7 z^5+a^7 z^3-a^5 z^7-2 a^5 z^5-a^5 z^3-a^3 z^7-2 a^3 z^5-a^3 z^3+a^3 z^{-1} +a z^5+a z^3-a z-a z^{-1} } (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 a^{11} z^5+5 a^{10} z^6-3 a^{10} z^4+12 a^9 z^7-14 a^9 z^5+5 a^9 z^3+17 a^8 z^8-25 a^8 z^6+13 a^8 z^4-2 a^8 z^2+13 a^7 z^9-7 a^7 z^7-14 a^7 z^5+8 a^7 z^3+4 a^6 z^{10}+26 a^6 z^8-64 a^6 z^6+38 a^6 z^4-6 a^6 z^2+23 a^5 z^9-34 a^5 z^7+a^5 z^5+6 a^5 z^3+4 a^4 z^{10}+19 a^4 z^8-54 a^4 z^6+34 a^4 z^4-6 a^4 z^2+10 a^3 z^9-10 a^3 z^7-9 a^3 z^5+8 a^3 z^3+a^3 z-a^3 z^{-1} +10 a^2 z^8-19 a^2 z^6+11 a^2 z^4-2 a^2 z^2+a^2+5 a z^7-9 a z^5+5 a z^3+a z-a z^{-1} +z^6-z^4} (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

4

0

7

1

-6

-2

13

4

9

-4

14

8

-6

17

12

5

-8

15

0

-10

12

16

-4

-12

8

15

7

-14

4

12

-8

-16

1

8

7

-18

4

-4

-20

1

Integral Khovanov Homology

(db, data source)

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 \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}}	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 i=-4}	$i=-2$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-5$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-4$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=-3$	${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{15}$	${\mathbb {Z} }^{15}$
$r=-2$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{16}$	${\mathbb {Z} }^{17}$
$r=-1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=0$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{13}$
$r=1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=3$	${\mathbb {Z} }_{2}$	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}}