L6a2

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L6a1.gif

L6a1

L6a3.gif

L6a3

Contents

L6a2.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L6a2 at Knotilus!

L6a2 is 6^2_2 in the Rolfsen table of links.

Mongolian ornament (4 crossings are unnecessary)

Link Presentations

[edit Notes on L6a2's Link Presentations]

Planar diagram presentation X8192 X12,5,7,6 X10,3,11,4 X4,11,5,12 X2738 X6,9,1,10
Gauss code {1, -5, 3, -4, 2, -6}, {5, -1, 6, -3, 4, -2}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation L6a2 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^2 (-v)-u v^2+u v-u-v}{u v} (db)
Jones polynomial -\frac{1}{q^{3/2}}+\frac{1}{q^{5/2}}-\frac{2}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{2}{q^{11/2}}+\frac{1}{q^{13/2}}-\frac{1}{q^{15/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^7 z+a^7 z^{-1} -a^5 z^3-2 a^5 z-a^5 z^{-1} -a^3 z^3-2 a^3 z (db)
Kauffman polynomial -z^3 a^9+2 z a^9-z^4 a^8+z^2 a^8-z^5 a^7+2 z^3 a^7-3 z a^7+a^7 z^{-1} -2 z^4 a^6+2 z^2 a^6-a^6-z^5 a^5+2 z^3 a^5-3 z a^5+a^5 z^{-1} -z^4 a^4+z^2 a^4-z^3 a^3+2 z a^3 (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 \
-6-5-4-3-2-10χ
-2      11
-4     110
-6    1  1
-8   11  0
-10  11   0
-12  1    1
-1411     0
-161      1
Integral Khovanov Homology

(db, data source)

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

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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