L11n426

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L11n425.gif

L11n425

L11n427.gif

L11n427

Contents

L11n426.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n426 at Knotilus!


Link Presentations

[edit Notes on L11n426's Link Presentations]

Planar diagram presentation X8192 X16,8,17,7 X14,6,15,5 X3,10,4,11 X4,14,5,13 X17,2,18,3 X9,19,10,18 X12,21,7,22 X22,11,13,12 X20,16,21,15 X6,19,1,20
Gauss code {1, 6, -4, -5, 3, -11}, {2, -1, -7, 4, 9, -8}, {5, -3, 10, -2, -6, 7, 11, -10, 8, -9}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation L11n426 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-t(2))^2 (t(3)-1)}{t(1) t(2) \sqrt{t(3)}} (db)
Jones polynomial  q^{-5} - q^{-4} -q^3+ q^{-3} +q^2+ q^{-1} +2 (db)
Signature 1 (db)
HOMFLY-PT polynomial a^4 z^2+a^4 z^{-2} +2 a^4-a^2 z^4-5 a^2 z^2-2 a^2 z^{-2} -z^2 a^{-2} -7 a^2-2 a^{-2} +z^4+5 z^2+ z^{-2} +7 (db)
Kauffman polynomial a^4 z^8-7 a^4 z^6+15 a^4 z^4-12 a^4 z^2-a^4 z^{-2} +5 a^4+a^3 z^9-7 a^3 z^7+14 a^3 z^5-7 a^3 z^3+z^3 a^{-3} -3 a^3 z-2 z a^{-3} +2 a^3 z^{-1} +2 a^2 z^8-15 a^2 z^6+35 a^2 z^4+z^4 a^{-2} -32 a^2 z^2-4 z^2 a^{-2} -2 a^2 z^{-2} +13 a^2+4 a^{-2} +a z^9-7 a z^7+13 a z^5-z^5 a^{-1} -3 a z^3+5 z^3 a^{-1} -7 a z-6 z a^{-1} +2 a z^{-1} +z^8-8 z^6+21 z^4-24 z^2- z^{-2} +13 (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 \
-6-5-4-3-2-10123χ
7         1-1
5          0
3      111 1
1      2   2
-1    113   3
-3   12     1
-5   12     1
-7 11       0
-9          0
-111         1
Integral Khovanov Homology

(db, data source)

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

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

See/edit the Link_Splice_Base (expert).

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L11n425

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L11n427