L11n406

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L11n405.gif

L11n405

L11n407.gif

L11n407

Contents

L11n406.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n406 at Knotilus!


Link Presentations

[edit Notes on L11n406's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 X11,22,12,19 X10,4,11,3 X5,21,6,20 X21,5,22,18 X19,12,20,13 X14,9,15,10 X2,14,3,13 X8,15,9,16
Gauss code {1, -10, 5, -3}, {-8, 6, -7, 4}, {-6, -1, 2, -11, 9, -5, -4, 8, 10, -9, 11, -2, 3, 7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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A Morse Link Presentation L11n406 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) 0 (db)
Jones polynomial q^3-q^2+2 q+ q^{-1} +2 q^{-4} - q^{-5} + q^{-6} - q^{-7} (db)
Signature -1 (db)
HOMFLY-PT polynomial -z^2 a^6-2 a^6+z^4 a^4+4 z^2 a^4+4 a^4+a^2 z^{-2} -z^4-4 z^2-2 z^{-2} -4+z^2 a^{-2} + a^{-2} z^{-2} +2 a^{-2} (db)
Kauffman polynomial a^7 z^7-6 a^7 z^5+10 a^7 z^3-4 a^7 z+a^6 z^8-6 a^6 z^6+11 a^6 z^4-10 a^6 z^2+4 a^6+2 a^5 z^7-12 a^5 z^5+18 a^5 z^3-8 a^5 z+2 a^4 z^8-14 a^4 z^6+30 a^4 z^4-28 a^4 z^2+8 a^4+a^3 z^9-6 a^3 z^7+8 a^3 z^5-2 a^3 z^3+2 a^2 z^8-13 a^2 z^6+z^6 a^{-2} +22 a^2 z^4-5 z^4 a^{-2} -12 a^2 z^2+6 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -4 a^{-2} +a z^9-6 a z^7+z^7 a^{-1} +10 a z^5-4 z^5 a^{-1} -10 a z^3+8 a z+4 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +z^8-4 z^6-2 z^4+12 z^2+2 z^{-2} -7 (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 \
-7-6-5-4-3-2-101234χ
7           11
5            0
3         21 1
1       31   2
-1      252   1
-3     122    1
-5    121     0
-7   222      2
-9   1        1
-11 121        0
-13            0
-151           -1
Integral Khovanov Homology

(db, data source)

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

See/edit the Link_Splice_Base (expert).

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L11n405.gif

L11n405

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L11n407