L7a3

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L7a2.gif

L7a2

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L7a4

Contents

L7a3.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L7a3 at Knotilus!

L7a3 is 7^2_4 in the Rolfsen table of links.


Link Presentations

[edit Notes on L7a3's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X12,8,13,7 X14,10,5,9 X8,14,9,13 X2536 X4,12,1,11
Gauss code {1, -6, 2, -7}, {6, -1, 3, -5, 4, -2, 7, -3, 5, -4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation L7a3 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) \left(t(2)^2+1\right)}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial 3 q^{9/2}-3 q^{7/2}+2 q^{5/2}-3 q^{3/2}+q^{13/2}-2 q^{11/2}+\sqrt{q}-\frac{1}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^3 a^{-5} +2 z a^{-5} + a^{-5} z^{-1} -z^5 a^{-3} -4 z^3 a^{-3} -5 z a^{-3} -3 a^{-3} z^{-1} +z^3 a^{-1} +3 z a^{-1} +2 a^{-1} z^{-1} (db)
Kauffman polynomial z^2 a^{-8} +2 z^3 a^{-7} +3 z^4 a^{-6} -3 z^2 a^{-6} + a^{-6} +3 z^5 a^{-5} -6 z^3 a^{-5} +4 z a^{-5} - a^{-5} z^{-1} +z^6 a^{-4} +z^4 a^{-4} -6 z^2 a^{-4} +3 a^{-4} +4 z^5 a^{-3} -12 z^3 a^{-3} +9 z a^{-3} -3 a^{-3} z^{-1} +z^6 a^{-2} -2 z^4 a^{-2} -2 z^2 a^{-2} +3 a^{-2} +z^5 a^{-1} -4 z^3 a^{-1} +5 z a^{-1} -2 a^{-1} z^{-1} (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 \
-2-1012345χ
14       1-1
12      1 1
10     21 -1
8    11  0
6   12   1
4  21    1
2 13     2
0        0
-21       1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=2 i=4
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{3} {\mathbb Z}^{2}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\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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L7a2

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L7a4