L11n408

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L11n407.gif

L11n407

L11n409.gif

L11n409

Contents

L11n408.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n408 at Knotilus!


Link Presentations

[edit Notes on L11n408's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X7,16,8,17 X17,19,18,22 X11,20,12,21 X19,10,20,11 X21,5,22,18 X9,14,10,15 X15,8,16,9 X2536 X4,14,1,13
Gauss code {1, -10, 2, -11}, {-6, 5, -7, 4}, {10, -1, -3, 9, -8, 6, -5, -2, 11, 8, -9, 3, -4, 7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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A Morse Link Presentation L11n408 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) (w-1)}{\sqrt{u} \sqrt{v} \sqrt{w}} (db)
Jones polynomial q^3-q^2+q+2- q^{-1} +3 q^{-2} -3 q^{-3} +4 q^{-4} -3 q^{-5} +2 q^{-6} - q^{-7} (db)
Signature 0 (db)
HOMFLY-PT polynomial a^6 \left(-z^2\right)-a^6+a^4 z^4+2 a^4 z^2+a^4+a^2 z^4+3 a^2 z^2+a^2 z^{-2} +z^2 a^{-2} + a^{-2} z^{-2} +3 a^2+2 a^{-2} -z^4-5 z^2-2 z^{-2} -5 (db)
Kauffman polynomial a^5 z^9+a^3 z^9+2 a^6 z^8+3 a^4 z^8+a^2 z^8+a^7 z^7-3 a^5 z^7-5 a^3 z^7+z^7 a^{-1} -10 a^6 z^6-15 a^4 z^6-5 a^2 z^6+z^6 a^{-2} +z^6-5 a^7 z^5-2 a^5 z^5+7 a^3 z^5-a z^5-5 z^5 a^{-1} +14 a^6 z^4+21 a^4 z^4+2 a^2 z^4-5 z^4 a^{-2} -10 z^4+7 a^7 z^3+6 a^5 z^3-7 a^3 z^3-3 a z^3+3 z^3 a^{-1} -7 a^6 z^2-13 a^4 z^2+7 a^2 z^2+6 z^2 a^{-2} +19 z^2-2 a^7 z-3 a^5 z+3 a^3 z+7 a z+3 z a^{-1} +2 a^6+2 a^4-6 a^2-4 a^{-2} -9-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2} (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         11 0
1       41   3
-1      241   1
-3     222    2
-5    22      0
-7   221      1
-9  12        1
-11 12         -1
-13 1          1
-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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z} {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}_2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{2} {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}^{4}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}_2 {\mathbb Z}
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

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

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L11n407.gif

L11n407

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L11n409