L8a7

From Knot Atlas
Jump to: navigation, search

L8a6.gif

L8a6

L8a8.gif

L8a8

Contents

L8a7.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L8a7 at Knotilus!

L8a7 is 8^2_{14} in the Rolfsen table of links.


Link Presentations

[edit Notes on L8a7's Link Presentations]

Planar diagram presentation X6172 X14,7,15,8 X4,15,1,16 X10,5,11,6 X12,3,13,4 X16,11,5,12 X2,9,3,10 X8,13,9,14
Gauss code {1, -7, 5, -3}, {4, -1, 2, -8, 7, -4, 6, -5, 8, -2, 3, -6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L8a7 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(2)^3+4 t(1) t(2)^2-4 t(2)^2-4 t(1) t(2)+4 t(2)+t(1)}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial \frac{5}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{4}{q^{15/2}}+\frac{6}{q^{13/2}}-\frac{7}{q^{11/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^9 z-a^9 z^{-1} -a^7 z^3+2 a^7 z+3 a^7 z^{-1} -3 a^5 z^3-5 a^5 z-2 a^5 z^{-1} -a^3 z^3 (db)
Kauffman polynomial a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-8 a^{10} z^4+5 a^{10} z^2+a^{10}+2 a^9 z^7-a^9 z^5-5 a^9 z^3+2 a^9 z-a^9 z^{-1} +8 a^8 z^6-16 a^8 z^4+4 a^8 z^2+3 a^8+2 a^7 z^7+4 a^7 z^5-12 a^7 z^3+6 a^7 z-3 a^7 z^{-1} +5 a^6 z^6-5 a^6 z^4-a^6 z^2+3 a^6+6 a^5 z^5-8 a^5 z^3+5 a^5 z-2 a^5 z^{-1} +3 a^4 z^4+a^3 z^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 \
-8-7-6-5-4-3-2-10χ
-2        11
-4       31-2
-6      3  3
-8     23  1
-10    53   2
-12   23    1
-14  24     -2
-16 12      1
-18 2       -2
-201        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}_2^{3} {\mathbb Z}^{3}
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).

Back to the top.

L8a6.gif

L8a6

L8a8.gif

L8a8