L11a488

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,8,13,7 X_16,10,17,9 X_22,18,19,17 X_20,14,21,13 X_14,20,15,19 X_18,22,5,21 X_8,16,9,15 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {7, -6, 8, -5}, {10, -1, 3, -9, 4, -2, 11, -3, 6, -7, 9, -4, 5, -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} , $w$ , ...)	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) (w-1)^2 \left(2 v w^2-v w+v+w^3-w^2+2 w\right)}{\sqrt{u} \sqrt{v} w^{5/2}}} (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^{11}+3 q^{10}-8 q^9+14 q^8-18 q^7+21 q^6-20 q^5+19 q^4-12 q^3+8 q^2-3 q+1} (db)
Signature	4 (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^2 a^{-10} - a^{-10} z^{-2} -2 a^{-10} +3 z^4 a^{-8} +8 z^2 a^{-8} +4 a^{-8} z^{-2} +9 a^{-8} -2 z^6 a^{-6} -7 z^4 a^{-6} -12 z^2 a^{-6} -5 a^{-6} z^{-2} -13 a^{-6} -z^6 a^{-4} -z^4 a^{-4} +3 z^2 a^{-4} +2 a^{-4} z^{-2} +5 a^{-4} +z^4 a^{-2} +2 z^2 a^{-2} + a^{-2} } (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^{10} a^{-6} +z^{10} a^{-8} +4 z^9 a^{-5} +9 z^9 a^{-7} +5 z^9 a^{-9} +5 z^8 a^{-4} +16 z^8 a^{-6} +19 z^8 a^{-8} +8 z^8 a^{-10} +3 z^7 a^{-3} -5 z^7 a^{-7} +4 z^7 a^{-9} +6 z^7 a^{-11} +z^6 a^{-2} -11 z^6 a^{-4} -51 z^6 a^{-6} -56 z^6 a^{-8} -14 z^6 a^{-10} +3 z^6 a^{-12} -7 z^5 a^{-3} -18 z^5 a^{-5} -29 z^5 a^{-7} -28 z^5 a^{-9} -9 z^5 a^{-11} +z^5 a^{-13} -3 z^4 a^{-2} +10 z^4 a^{-4} +64 z^4 a^{-6} +71 z^4 a^{-8} +16 z^4 a^{-10} -4 z^4 a^{-12} +4 z^3 a^{-3} +24 z^3 a^{-5} +50 z^3 a^{-7} +39 z^3 a^{-9} +7 z^3 a^{-11} -2 z^3 a^{-13} +3 z^2 a^{-2} -11 z^2 a^{-4} -48 z^2 a^{-6} -48 z^2 a^{-8} -13 z^2 a^{-10} +z^2 a^{-12} -17 z a^{-5} -35 z a^{-7} -23 z a^{-9} -4 z a^{-11} +z a^{-13} - a^{-2} +8 a^{-4} +22 a^{-6} +19 a^{-8} +5 a^{-10} +5 a^{-5} z^{-1} +9 a^{-7} z^{-1} +5 a^{-9} z^{-1} + a^{-11} z^{-1} -2 a^{-4} z^{-2} -5 a^{-6} z^{-2} -4 a^{-8} z^{-2} - a^{-10} z^{-2} } (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		\

-2

-1

0

1

2

3

4

5

6

7

8

9

χ

23

1

-1

21

2

19

6

1

-5

17

8

2

6

15

10

6

-4

13

11

8

3

11

12

1

9

8

9

-1

7

4

11

7

5

4

8

-4

3

1

6

5

1

2

-2

-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}}	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=3}	$i=5$
$r=-2$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{4}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=4$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$	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}}
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}^{8}\oplus{\mathbb Z}_2^{10}}	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}^{10}}
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}^{6}\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}}
$r=7$	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}\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}}
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=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}\oplus{\mathbb Z}_2^{2}}	${\mathbb {Z} }^{2}$
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 r=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}_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}}