L11a180

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L11a179.gif

L11a179

L11a181.gif

L11a181

L11a180.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a180's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X14,6,15,5 X16,8,17,7 X22,18,7,17 X12,15,13,16 X20,10,21,9 X18,11,19,12 X6,14,1,13 X4,20,5,19 X2,21,3,22
Gauss code {1, -11, 2, -10, 3, -9}, {4, -1, 7, -2, 8, -6, 9, -3, 6, -4, 5, -8, 10, -7, 11, -5}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a180 ML.gif

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{t(1)^2 t(2)^4-2 t(1) t(2)^4+t(2)^4-5 t(1)^2 t(2)^3+11 t(1) t(2)^3-6 t(2)^3+9 t(1)^2 t(2)^2-19 t(1) t(2)^2+9 t(2)^2-6 t(1)^2 t(2)+11 t(1) t(2)-5 t(2)+t(1)^2-2 t(1)+1}{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^{15/2}-4 q^{13/2}+10 q^{11/2}-18 q^{9/2}+24 q^{7/2}-29 q^{5/2}+29 q^{3/2}-26 \sqrt{q}+\frac{19}{\sqrt{q}}-\frac{12}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{1}{q^{7/2}}} (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^7 a^{-1} +a z^5-3 z^5 a^{-1} +3 z^5 a^{-3} +a z^3-5 z^3 a^{-1} +5 z^3 a^{-3} -3 z^3 a^{-5} +a z-z a^{-1} +2 z a^{-3} -2 z a^{-5} +z a^{-7} + a^{-1} z^{-1} - a^{-3} 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} +6 z^3 a^{-7} -2 z a^{-7} +8 z^8 a^{-6} -16 z^6 a^{-6} +12 z^4 a^{-6} -4 z^2 a^{-6} +8 z^9 a^{-5} -7 z^7 a^{-5} -10 z^5 a^{-5} +12 z^3 a^{-5} -3 z a^{-5} +3 z^{10} a^{-4} +19 z^8 a^{-4} -54 z^6 a^{-4} +41 z^4 a^{-4} -10 z^2 a^{-4} +19 z^9 a^{-3} -23 z^7 a^{-3} +a^3 z^5-12 z^5 a^{-3} +15 z^3 a^{-3} - a^{-3} z^{-1} +3 z^{10} a^{-2} +27 z^8 a^{-2} +5 a^2 z^6-65 z^6 a^{-2} -3 a^2 z^4+41 z^4 a^{-2} -8 z^2 a^{-2} + a^{-2} +11 z^9 a^{-1} +12 a z^7-15 a z^5-26 z^5 a^{-1} +6 a z^3+15 z^3 a^{-1} -a z- a^{-1} z^{-1} +16 z^8-23 z^6+11 z^4-3 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 (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 \
-4-3-2-101234567χ
16           1-1
14          3 3
12         71 -6
10        113  8
8       137   -6
6      1611    5
4     1414     0
2    1215      -3
0   815       7
-2  411        -7
-4 18         7
-6 4          -4
-81           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}
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} 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}}
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}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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