L8a2

From Knot Atlas
Jump to: navigation, search

L8a1.gif

L8a1

L8a3.gif

L8a3

Contents

L8a2.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L8a2 at Knotilus!

L8a2 is 8^2_{10} in the Rolfsen table of links.


Link Presentations

[edit Notes on L8a2's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X12,8,13,7 X16,14,5,13 X14,9,15,10 X8,15,9,16 X2536 X4,12,1,11
Gauss code {1, -7, 2, -8}, {7, -1, 3, -6, 5, -2, 8, -3, 4, -5, 6, -4}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L8a2 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1)^3}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial -q^{9/2}+3 q^{7/2}-5 q^{5/2}+5 q^{3/2}-6 \sqrt{q}+\frac{5}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{7/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z^5 a^{-1} -2 a z^3+3 z^3 a^{-1} -z^3 a^{-3} +a^3 z-4 a z+4 z a^{-1} -z a^{-3} +a^3 z^{-1} -2 a z^{-1} +2 a^{-1} z^{-1} - a^{-3} z^{-1} (db)
Kauffman polynomial z^3 a^{-5} +3 z^4 a^{-4} -z^2 a^{-4} +a^3 z^5+5 z^5 a^{-3} -3 a^3 z^3-6 z^3 a^{-3} +3 a^3 z+3 z a^{-3} -a^3 z^{-1} - a^{-3} z^{-1} +2 a^2 z^6+4 z^6 a^{-2} -5 a^2 z^4-4 z^4 a^{-2} +3 a^2 z^2+z^2 a^{-2} +a z^7+z^7 a^{-1} +3 a z^5+7 z^5 a^{-1} -13 a z^3-17 z^3 a^{-1} +10 a z+10 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +6 z^6-12 z^4+5 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 \
-4-3-2-101234χ
10        11
8       2 -2
6      31 2
4     22  0
2    43   1
0   34    1
-2  12     -1
-4 13      2
-6 1       -1
-81        1
Integral Khovanov Homology

(db, data source)

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

Back to the top.

L8a1.gif

L8a1

L8a3.gif

L8a3