L10a30

From Knot Atlas
Jump to: navigation, search

L10a29.gif

L10a29

L10a31.gif

L10a31

Contents

L10a30.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a30 at Knotilus!


Link Presentations

[edit Notes on L10a30's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X14,8,15,7 X18,10,19,9 X20,16,5,15 X16,20,17,19 X8,18,9,17 X10,14,11,13 X2536 X4,11,1,12
Gauss code {1, -9, 2, -10}, {9, -1, 3, -7, 4, -8, 10, -2, 8, -3, 5, -6, 7, -4, 6, -5}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10a30 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 t(1) t(2)^3-4 t(2)^3-6 t(1) t(2)^2+7 t(2)^2+7 t(1) t(2)-6 t(2)-4 t(1)+2}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial -q^{13/2}+4 q^{11/2}-7 q^{9/2}+10 q^{7/2}-12 q^{5/2}+13 q^{3/2}-12 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{7/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^3 a^{-5} +z^5 a^{-3} +z^3 a^{-3} +a^3 z+z a^{-3} +a^3 z^{-1} + a^{-3} z^{-1} +z^5 a^{-1} -2 a z^3-2 a z-2 z a^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial z^5 a^{-7} -z^3 a^{-7} +4 z^6 a^{-6} -7 z^4 a^{-6} +2 z^2 a^{-6} +6 z^7 a^{-5} -11 z^5 a^{-5} +4 z^3 a^{-5} +4 z^8 a^{-4} -z^6 a^{-4} -9 z^4 a^{-4} +7 z^2 a^{-4} -2 a^{-4} +z^9 a^{-3} +9 z^7 a^{-3} +a^3 z^5-18 z^5 a^{-3} -3 a^3 z^3+8 z^3 a^{-3} +3 a^3 z-z a^{-3} -a^3 z^{-1} + a^{-3} z^{-1} +6 z^8 a^{-2} +2 a^2 z^6-3 z^6 a^{-2} -3 a^2 z^4-12 z^4 a^{-2} +16 z^2 a^{-2} +a^2-5 a^{-2} +z^9 a^{-1} +3 a z^7+6 z^7 a^{-1} -4 a z^5-11 z^5 a^{-1} +2 a z^3+8 z^3 a^{-1} -a z-5 z a^{-1} +2 a^{-1} z^{-1} +2 z^8+4 z^6-13 z^4+11 z^2-3 (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 \
-4-3-2-10123456χ
14          11
12         3 -3
10        41 3
8       63  -3
6      64   2
4     76    -1
2    56     -1
0   48      4
-2  24       -2
-4  4        4
-612         -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} {\mathbb Z}
r=-3 {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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.

L10a29.gif

L10a29

L10a31.gif

L10a31