L11a146

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L11a145.gif

L11a145

L11a147.gif

L11a147

Contents

L11a146.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a146 at Knotilus!


Link Presentations

[edit Notes on L11a146's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X16,8,17,7 X20,14,21,13 X22,16,7,15 X6,19,1,20 X18,11,19,12 X12,6,13,5 X14,22,15,21 X4,18,5,17
Gauss code {1, -2, 3, -11, 9, -7}, {4, -1, 2, -3, 8, -9, 5, -10, 6, -4, 11, -8, 7, -5, 10, -6}
A Braid Representative
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gif
A Morse Link Presentation L11a146 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1)^2 t(2)^6-t(1) t(2)^6-2 t(1)^2 t(2)^5+3 t(1) t(2)^5-t(2)^5+2 t(1)^2 t(2)^4-5 t(1) t(2)^4+2 t(2)^4-2 t(1)^2 t(2)^3+5 t(1) t(2)^3-2 t(2)^3+2 t(1)^2 t(2)^2-5 t(1) t(2)^2+2 t(2)^2-t(1)^2 t(2)+3 t(1) t(2)-2 t(2)-t(1)+1}{t(1) t(2)^3} (db)
Jones polynomial -q^{13/2}+3 q^{11/2}-5 q^{9/2}+9 q^{7/2}-12 q^{5/2}+13 q^{3/2}-14 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{9}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-3} -5 z^5 a^{-3} -7 z^3 a^{-3} -z a^{-3} + a^{-3} z^{-1} +z^9 a^{-1} -a z^7+7 z^7 a^{-1} -5 a z^5+17 z^5 a^{-1} -7 a z^3+15 z^3 a^{-1} -a z+z a^{-1} +2 a z^{-1} -3 a^{-1} z^{-1} (db)
Kauffman polynomial z^5 a^{-7} -2 z^3 a^{-7} +3 z^6 a^{-6} -7 z^4 a^{-6} +3 z^2 a^{-6} +4 z^7 a^{-5} -8 z^5 a^{-5} +3 z^3 a^{-5} +4 z^8 a^{-4} +a^4 z^6-7 z^6 a^{-4} -3 a^4 z^4+2 z^4 a^{-4} +a^4 z^2+3 z^2 a^{-4} - a^{-4} +4 z^9 a^{-3} +3 a^3 z^7-12 z^7 a^{-3} -10 a^3 z^5+20 z^5 a^{-3} +7 a^3 z^3-13 z^3 a^{-3} -a^3 z+2 z a^{-3} + a^{-3} z^{-1} +2 z^{10} a^{-2} +4 a^2 z^8-3 z^8 a^{-2} -12 a^2 z^6+7 a^2 z^4+7 z^4 a^{-2} -3 a^{-2} +4 a z^9+8 z^9 a^{-1} -13 a z^7-32 z^7 a^{-1} +14 a z^5+53 z^5 a^{-1} -6 a z^3-31 z^3 a^{-1} -2 a z+z a^{-1} +2 a z^{-1} +3 a^{-1} z^{-1} +2 z^{10}-3 z^8-3 z^6+8 z^4-z^2-3 (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r).   
\ r
  \  
j \
-5-4-3-2-10123456χ
14           11
12          2 -2
10         31 2
8        62  -4
6       63   3
4      76    -1
2     76     1
0    58      3
-2   46       -2
-4  26        4
-6 13         -2
-8 2          2
-101           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\mathbb Z}_2 {\mathbb Z}

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).

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L11a145

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L11a147