L8a20

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L8a19.gif

L8a19

L8a21.gif

L8a21

Contents

L8a20.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L8a20 at Knotilus!

L8a20 is 8^3_{4} in the Rolfsen table of links.

Depiction obtained with knotilus

Link Presentations

[edit Notes on L8a20's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X16,12,13,11 X14,8,15,7 X8,14,9,13 X12,16,5,15 X2536 X4,9,1,10
Gauss code {1, -7, 2, -8}, {5, -4, 6, -3}, {7, -1, 4, -5, 8, -2, 3, -6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L8a20 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(w-1) \left(u v w-u v-2 u w-2 v w-w^2+w\right)}{\sqrt{u} \sqrt{v} w^{3/2}} (db)
Jones polynomial q^4-2 q^3+5 q^2-5 q+6-5 q^{-1} +5 q^{-2} -2 q^{-3} + q^{-4} (db)
Signature 0 (db)
HOMFLY-PT polynomial a^4+ a^{-4} -2 a^2 z^2+a^2 z^{-2} -2 z^2 a^{-2} + a^{-2} z^{-2} +z^4-2 z^{-2} -2 (db)
Kauffman polynomial a^4 z^4+z^4 a^{-4} -2 a^4 z^2-2 z^2 a^{-4} +a^4+ a^{-4} +2 a^3 z^5+2 z^5 a^{-3} -2 a^3 z^3-2 z^3 a^{-3} +3 a^2 z^6+3 z^6 a^{-2} -5 a^2 z^4-5 z^4 a^{-2} +5 a^2 z^2+5 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -4 a^2-4 a^{-2} +a z^7+z^7 a^{-1} +5 a z^5+5 z^5 a^{-1} -12 a z^3-12 z^3 a^{-1} +8 a z+8 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +6 z^6-12 z^4+14 z^2+2 z^{-2} -9 (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 \
-4-3-2-101234χ
9        11
7       21-1
5      3  3
3     22  0
1    43   1
-1   34    1
-3  22     0
-5  3      3
-712       -1
-91        1
Integral Khovanov Homology

(db, data source)

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

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

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L8a19

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