L7a6

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L7a5.gif

L7a5

L7a7.gif

L7a7

Contents

L7a6.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L7a6 at Knotilus!

L7a6 is 7^2_1 in the Rolfsen table of links.


Link Presentations

[edit Notes on L7a6's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X14,10,7,9 X12,6,13,5 X2738 X4,12,5,11 X6,14,1,13
Gauss code {1, -5, 2, -6, 4, -7}, {5, -1, 3, -2, 6, -4, 7, -3}
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation L7a6 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(2)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(1)+t(2) t(1)-t(1)-t(2)+1}{t(1) t(2)} (db)
Jones polynomial q^{13/2}-2 q^{11/2}+2 q^{9/2}-3 q^{7/2}+2 q^{5/2}-2 q^{3/2}+\sqrt{q}-\frac{1}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial -z^5 a^{-3} +z^3 a^{-1} -4 z^3 a^{-3} +z^3 a^{-5} +3 z a^{-1} -4 z a^{-3} +2 z a^{-5} + a^{-1} z^{-1} - a^{-3} z^{-1} (db)
Kauffman polynomial z^2 a^{-8} +2 z^3 a^{-7} -z a^{-7} +2 z^4 a^{-6} -z^2 a^{-6} +2 z^5 a^{-5} -3 z^3 a^{-5} +z a^{-5} +z^6 a^{-4} -z^4 a^{-4} -z^2 a^{-4} +3 z^5 a^{-3} -9 z^3 a^{-3} +6 z a^{-3} - a^{-3} z^{-1} +z^6 a^{-2} -3 z^4 a^{-2} +z^2 a^{-2} + a^{-2} +z^5 a^{-1} -4 z^3 a^{-1} +4 z a^{-1} - 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     11 0
8    21  1
6   12   1
4  11    0
2 12     1
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}^{2} {\mathbb Z}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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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L7a5

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L7a7