L7a1

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L6n1.gif

L6n1

L7a2.gif

L7a2

Contents

L7a1.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L7a1 at Knotilus!

L7a1 is 7^2_6 in the Rolfsen table of links.


Link Presentations

[edit Notes on L7a1's Link Presentations]

Planar diagram presentation X6172 X12,7,13,8 X4,13,1,14 X10,6,11,5 X8493 X14,10,5,9 X2,12,3,11
Gauss code {1, -7, 5, -3}, {4, -1, 2, -5, 6, -4, 7, -2, 3, -6}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gif
A Morse Link Presentation L7a1 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(v^2-v+1\right)}{\sqrt{u} v^{3/2}} (db)
Jones polynomial -q^{9/2}+3 q^{7/2}-4 q^{5/2}+\frac{1}{q^{5/2}}+4 q^{3/2}-\frac{3}{q^{3/2}}-5 \sqrt{q}+\frac{3}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^3 a^{-3} -z a^{-3} +z^5 a^{-1} -a z^3+3 z^3 a^{-1} -a z+2 z a^{-1} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial z^3 a^{-5} +3 z^4 a^{-4} -2 z^2 a^{-4} +4 z^5 a^{-3} -5 z^3 a^{-3} +2 z a^{-3} +2 z^6 a^{-2} +a^2 z^4+z^4 a^{-2} -a^2 z^2-3 z^2 a^{-2} +3 a z^5+7 z^5 a^{-1} -6 a z^3-12 z^3 a^{-1} +2 a z+4 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +2 z^6-z^4-2 z^2-1 (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 \
-3-2-101234χ
10       11
8      2 -2
6     21 1
4    22  0
2   32   1
0  24    2
-2 11     0
-4 2      2
-61       -1
Integral Khovanov Homology

(db, data source)

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

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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L6n1.gif

L6n1

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L7a2