L11a460

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L11a459.gif

L11a459

L11a461.gif

L11a461

Contents

L11a460.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a460's Link Presentations]

Planar diagram presentation X6172 X14,4,15,3 X22,10,13,9 X20,8,21,7 X8,14,9,13 X18,15,19,16 X16,6,17,5 X12,18,5,17 X10,22,11,21 X2,11,3,12 X4,20,1,19
Gauss code {1, -10, 2, -11}, {7, -1, 4, -5, 3, -9, 10, -8}, {5, -2, 6, -7, 8, -6, 11, -4, 9, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a460 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(3)^2 t(2)^3-t(3)^2 t(2)^3-2 t(1) t(3) t(2)^3+2 t(3) t(2)^3-t(2)^3+t(1) t(3)^3 t(2)^2-t(3)^3 t(2)^2-5 t(1) t(3)^2 t(2)^2+5 t(3)^2 t(2)^2-2 t(1) t(2)^2+6 t(1) t(3) t(2)^2-6 t(3) t(2)^2+2 t(2)^2-2 t(1) t(3)^3 t(2)+2 t(3)^3 t(2)+6 t(1) t(3)^2 t(2)-6 t(3)^2 t(2)+t(1) t(2)-5 t(1) t(3) t(2)+5 t(3) t(2)-t(2)+t(1) t(3)^3-2 t(1) t(3)^2+2 t(3)^2+t(1) t(3)-t(3)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} (db)
Jones polynomial q^9-3 q^8+8 q^7-12 q^6+19 q^5-22 q^4+23 q^3-20 q^2- q^{-2} +16 q+5 q^{-1} -10 (db)
Signature 2 (db)
HOMFLY-PT polynomial z^6 a^{-2} +z^6 a^{-4} +z^4 a^{-2} +z^4 a^{-4} -2 z^4 a^{-6} -z^4+z^2 a^{-4} -3 z^2 a^{-6} +z^2 a^{-8} + a^{-4} -3 a^{-6} + a^{-8} +1+ a^{-4} z^{-2} -2 a^{-6} z^{-2} + a^{-8} z^{-2} (db)
Kauffman polynomial z^6 a^{-10} -3 z^4 a^{-10} +2 z^2 a^{-10} +3 z^7 a^{-9} -7 z^5 a^{-9} +3 z^3 a^{-9} +6 z^8 a^{-8} -18 z^6 a^{-8} +24 z^4 a^{-8} -20 z^2 a^{-8} - a^{-8} z^{-2} +8 a^{-8} +5 z^9 a^{-7} -6 z^7 a^{-7} -7 z^5 a^{-7} +17 z^3 a^{-7} -11 z a^{-7} +2 a^{-7} z^{-1} +2 z^{10} a^{-6} +10 z^8 a^{-6} -36 z^6 a^{-6} +46 z^4 a^{-6} -31 z^2 a^{-6} -2 a^{-6} z^{-2} +13 a^{-6} +12 z^9 a^{-5} -19 z^7 a^{-5} +z^5 a^{-5} +17 z^3 a^{-5} -11 z a^{-5} +2 a^{-5} z^{-1} +2 z^{10} a^{-4} +15 z^8 a^{-4} -36 z^6 a^{-4} +26 z^4 a^{-4} -10 z^2 a^{-4} - a^{-4} z^{-2} +5 a^{-4} +7 z^9 a^{-3} -14 z^5 a^{-3} +6 z^3 a^{-3} +11 z^8 a^{-2} -14 z^6 a^{-2} +2 z^4 a^{-2} -z^2 a^{-2} +10 z^7 a^{-1} +a z^5-14 z^5 a^{-1} +3 z^3 a^{-1} +5 z^6-5 z^4+1 (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 \
-3-2-1012345678χ
19           11
17          31-2
15         5  5
13        73  -4
11       125   7
9      118    -3
7     1211     1
5    811      3
3   812       -4
1  410        6
-1 16         -5
-3 4          4
-51           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=7 {\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=8 {\mathbb Z} {\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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L11a459.gif

L11a459

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L11a461