L8a12

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L8a11.gif

L8a11

L8a13.gif

L8a13

Contents

L8a12.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L8a12 at Knotilus!

L8a12 is 8^2_{2} in the Rolfsen table of links.


Link Presentations

[edit Notes on L8a12's Link Presentations]

Planar diagram presentation X10,1,11,2 X16,7,9,8 X12,3,13,4 X14,5,15,6 X4,13,5,14 X6,15,7,16 X2,9,3,10 X8,11,1,12
Gauss code {1, -7, 3, -5, 4, -6, 2, -8}, {7, -1, 8, -3, 5, -4, 6, -2}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation L8a12 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(2)^2 t(1)^3-t(2)^3 t(1)^2+t(2)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(1)+t(2) t(1)-t(1)-t(2)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -\frac{1}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{2}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{2}{q^{17/2}}+\frac{1}{q^{19/2}}-\frac{1}{q^{21/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial a^9 z^3+3 a^9 z+a^9 z^{-1} -a^7 z^5-4 a^7 z^3-4 a^7 z-a^7 z^{-1} -a^5 z^5-4 a^5 z^3-3 a^5 z (db)
Kauffman polynomial -z^3 a^{13}+2 z a^{13}-z^4 a^{12}+z^2 a^{12}-z^5 a^{11}+z^3 a^{11}-z a^{11}-z^6 a^{10}+2 z^4 a^{10}-3 z^2 a^{10}-z^7 a^9+4 z^5 a^9-8 z^3 a^9+5 z a^9-a^9 z^{-1} -2 z^6 a^8+6 z^4 a^8-5 z^2 a^8+a^8-z^7 a^7+4 z^5 a^7-6 z^3 a^7+5 z a^7-a^7 z^{-1} -z^6 a^6+3 z^4 a^6-z^2 a^6-z^5 a^5+4 z^3 a^5-3 z a^5 (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 \
-8-7-6-5-4-3-2-10χ
-4        11
-6       110
-8      1  1
-10     11  0
-12    21   1
-14   11    0
-16  12     -1
-18  1      1
-2011       0
-221        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-6 i=-4
r=-8 {\mathbb Z} {\mathbb Z}
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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L8a11.gif

L8a11

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L8a13