L11n392

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_5,12,6,13 X₈₄₉₃ X_22,14,19,13 X_20,10,21,9 X_10,20,11,19 X_14,22,15,21 X_11,18,12,5 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {8, -7, 9, -6}, {-4, -1, 2, -5, 7, -8, -10, 4, 6, -9, 11, -2, 3, 10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(3)-1)^{2}(t(3)t(2)-2t(2)-2t(3)+1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-5}+3q^{4}+4q^{-4}-6q^{3}-9q^{-3}+13q^{2}+13q^{-2}-14q-16q^{-1}+17$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{4}z^{2}+2a^{-4}z^{-2}+2a^{-4}+2a^{2}z^{4}+z^{4}a^{-2}+a^{2}z^{2}-3z^{2}a^{-2}-a^{2}z^{-2}-5a^{-2}z^{-2}-2a^{2}-8a^{-2}-z^{6}-z^{4}+3z^{2}+4z^{-2}+8$ (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^5 z^5-a^5 z^3+4 a^4 z^6-5 a^4 z^4+6 z^4 a^{-4} +2 a^4 z^2-11 z^2 a^{-4} -2 a^{-4} z^{-2} +8 a^{-4} +8 a^3 z^7+3 z^7 a^{-3} -14 a^3 z^5-3 z^5 a^{-3} +9 a^3 z^3+10 z^3 a^{-3} -3 a^3 z-13 z a^{-3} +a^3 z^{-1} +5 a^{-3} z^{-1} +7 a^2 z^8+5 z^8 a^{-2} -6 a^2 z^6-8 z^6 a^{-2} -4 a^2 z^4+17 z^4 a^{-2} -24 z^2 a^{-2} -a^2 z^{-2} -5 a^{-2} z^{-2} +2 a^2+16 a^{-2} +2 a z^9+2 z^9 a^{-1} +15 a z^7+10 z^7 a^{-1} -38 a z^5-26 z^5 a^{-1} +29 a z^3+29 z^3 a^{-1} -14 a z-24 z a^{-1} +5 a z^{-1} +9 a^{-1} z^{-1} +12 z^8-18 z^6+12 z^4-15 z^2-4 z^{-2} +11} (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

j

, alternation over

r

).

\		r
	\
j		\

-5

-4

-3

-2

-1

0

1

2

3

4

χ

9

3

7

5

2

-3

5

8

1

7

3

6

5

-1

1

11

8

3

-1

7

8

1

-3

6

9

-3

-5

3

7

4

-7

1

6

-5

-9

3

-11

1

-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}}	$i=-1$	$i=1$
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=-5}	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}}
$r=-4$	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}\oplus{\mathbb Z}_2}	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
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=-2}	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$	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}^{6}}
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
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=0}	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$	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}^{11}}
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$	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}^{8}}
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{3}$