L11a79

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_16,8,17,7 X_22,18,5,17 X_18,14,19,13 X_14,22,15,21 X_20,10,21,9 X_8,16,9,15 X_10,20,11,19 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -8, 7, -9, 11, -2, 5, -6, 8, -3, 4, -5, 9, -7, 6, -4}

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{3 t(1) t(2)^3-7 t(2)^3-11 t(1) t(2)^2+14 t(2)^2+14 t(1) t(2)-11 t(2)-7 t(1)+3}{\sqrt{t(1)} t(2)^{3/2}}} (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 -15 q^{9/2}+20 q^{7/2}-\frac{1}{q^{7/2}}-23 q^{5/2}+\frac{3}{q^{5/2}}+22 q^{3/2}-\frac{8}{q^{3/2}}+q^{15/2}-4 q^{13/2}+10 q^{11/2}-20 \sqrt{q}+\frac{13}{\sqrt{q}}} (db)
Signature	1 (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 a^{-7} -3 z^3 a^{-5} -z a^{-5} + a^{-5} z^{-1} +2 z^5 a^{-3} +2 z^3 a^{-3} +a^3 z+z a^{-3} - a^{-3} z^{-1} +z^5 a^{-1} -2 a z^3-3 z^3 a^{-1} +a z-5 z a^{-1} +2 a z^{-1} -2 a^{-1} 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 z^6 a^{-8} -2 z^4 a^{-8} +z^2 a^{-8} +4 z^7 a^{-7} -8 z^5 a^{-7} +4 z^3 a^{-7} -z a^{-7} +8 z^8 a^{-6} -19 z^6 a^{-6} +15 z^4 a^{-6} -10 z^2 a^{-6} +4 a^{-6} +7 z^9 a^{-5} -9 z^7 a^{-5} -7 z^5 a^{-5} +9 z^3 a^{-5} -2 z a^{-5} - a^{-5} z^{-1} +2 z^{10} a^{-4} +16 z^8 a^{-4} -54 z^6 a^{-4} +59 z^4 a^{-4} -34 z^2 a^{-4} +9 a^{-4} +13 z^9 a^{-3} -21 z^7 a^{-3} +a^3 z^5+2 z^5 a^{-3} -2 a^3 z^3+9 z^3 a^{-3} +a^3 z-2 z a^{-3} - a^{-3} z^{-1} +2 z^{10} a^{-2} +15 z^8 a^{-2} +3 a^2 z^6-44 z^6 a^{-2} -4 a^2 z^4+46 z^4 a^{-2} +a^2 z^2-20 z^2 a^{-2} +4 a^{-2} +6 z^9 a^{-1} +6 a z^7-2 z^7 a^{-1} -10 a z^5-10 z^5 a^{-1} +10 a z^3+16 z^3 a^{-1} -7 a z-9 z a^{-1} +2 a z^{-1} +2 a^{-1} z^{-1} +7 z^8-7 z^6+4 z^2-2} (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

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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

16

1

-1

14

3

12

7

1

-6

10

8

3

5

8

12

7

-5

6

11

8

3

4

11

12

1

2

9

11

-2

0

5

12

7

-2

3

8

-5

-4

5

-6

1

3

-2

-8

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=0}	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=2}
$r=-4$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
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}	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}^{12}\oplus{\mathbb Z}_2^{11}}	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}}
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=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}^{8}\oplus{\mathbb Z}_2^{12}}	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}^{12}}
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=4}	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$	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}}
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}^{3}\oplus{\mathbb Z}_2^{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 {\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=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}\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=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 {\mathbb Z}_2}	${\mathbb {Z} }$