L8a15

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L8a14.gif

L8a14

L8a16.gif

L8a16

Contents

L8a15.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L8a15 at Knotilus!

L8a15 is 8^3_{3} in the Rolfsen table of links.


Link Presentations

[edit Notes on L8a15's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X16,12,9,11 X12,16,13,15 X2536 X4,9,1,10
Gauss code {1, -7, 2, -8}, {7, -1, 3, -4}, {8, -2, 5, -6, 4, -3, 6, -5}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L8a15 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u v w-2 u v-2 u w+2 u-2 v w+2 v+2 w-1}{\sqrt{u} \sqrt{v} \sqrt{w}} (db)
Jones polynomial  q^{-7} - q^{-6} +4 q^{-5} -4 q^{-4} +6 q^{-3} -4 q^{-2} +q+4 q^{-1} -3 (db)
Signature -2 (db)
HOMFLY-PT polynomial a^8 z^{-2} -2 a^6 z^{-2} -3 a^6+3 a^4 z^2+a^4 z^{-2} +3 a^4-a^2 z^4-a^2 z^2+z^2 (db)
Kauffman polynomial z^4 a^8-3 z^2 a^8-a^8 z^{-2} +3 a^8+z^5 a^7-3 z a^7+2 a^7 z^{-1} +z^6 a^6+2 z^4 a^6-6 z^2 a^6-2 a^6 z^{-2} +5 a^6+z^7 a^5+2 z^3 a^5-3 z a^5+2 a^5 z^{-1} +4 z^6 a^4-5 z^4 a^4-a^4 z^{-2} +3 a^4+z^7 a^3+2 z^5 a^3-3 z^3 a^3+3 z^6 a^2-5 z^4 a^2+2 z^2 a^2+3 z^5 a-5 z^3 a+z^4-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 \
-6-5-4-3-2-1012χ
3        11
1       2 -2
-1      21 1
-3     33  0
-5    31   2
-7   13    2
-9  33     0
-11 14      3
-13         0
-151        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{4} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\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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L8a14.gif

L8a14

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L8a16