L10a16

From Knot Atlas
Jump to: navigation, search

L10a15.gif

L10a15

L10a17.gif

L10a17

Contents

L10a16.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a16 at Knotilus!


Link Presentations

[edit Notes on L10a16's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X16,8,17,7 X20,11,5,12 X18,13,19,14 X14,17,15,18 X12,19,13,20 X8,16,9,15 X2536 X4,9,1,10
Gauss code {1, -9, 2, -10}, {9, -1, 3, -8, 10, -2, 4, -7, 5, -6, 8, -3, 6, -5, 7, -4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation L10a16 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 t(2)^3+3 t(1) t(2)^2-6 t(2)^2-6 t(1) t(2)+3 t(2)+2 t(1)}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial -\frac{6}{q^{9/2}}+\frac{7}{q^{7/2}}-\frac{7}{q^{5/2}}-q^{3/2}+\frac{5}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{5}{q^{11/2}}+2 \sqrt{q}-\frac{4}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^9 z^{-1} -3 a^7 z-2 a^7 z^{-1} +2 a^5 z^3+2 a^5 z+a^5 z^{-1} +2 a^3 z^3+a^3 z+a^3 z^{-1} +a z^3-a z-a z^{-1} -z a^{-1} (db)
Kauffman polynomial -z^7 a^9+5 z^5 a^9-8 z^3 a^9+5 z a^9-a^9 z^{-1} -2 z^8 a^8+9 z^6 a^8-11 z^4 a^8+4 z^2 a^8-a^8-z^9 a^7+15 z^5 a^7-26 z^3 a^7+13 z a^7-2 a^7 z^{-1} -5 z^8 a^6+20 z^6 a^6-23 z^4 a^6+11 z^2 a^6-3 a^6-z^9 a^5-2 z^7 a^5+16 z^5 a^5-18 z^3 a^5+8 z a^5-a^5 z^{-1} -3 z^8 a^4+8 z^6 a^4-8 z^4 a^4+7 z^2 a^4-2 a^4-3 z^7 a^3+3 z^5 a^3+4 z^3 a^3-4 z a^3+a^3 z^{-1} -3 z^6 a^2+2 z^4 a^2+z^2 a^2-a^2-3 z^5 a+3 z^3 a-3 z a+a z^{-1} -2 z^4+z^2-z^3 a^{-1} +z a^{-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 \
-8-7-6-5-4-3-2-1012χ
4          11
2         1 -1
0        31 2
-2       32  -1
-4      42   2
-6     33    0
-8    34     -1
-10   34      1
-12  12       -1
-14 13        2
-16 1         -1
-181          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\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.

L10a15.gif

L10a15

L10a17.gif

L10a17