L11n335

From Knot Atlas
Jump to: navigation, search

L11n334.gif

L11n334

L11n336.gif

L11n336

Contents

L11n335.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n335 at Knotilus!


Link Presentations

[edit Notes on L11n335's Link Presentations]

Planar diagram presentation X6172 X11,16,12,17 X8493 X2,18,3,17 X5,14,6,15 X18,7,19,8 X15,12,16,5 X20,14,21,13 X22,9,13,10 X10,21,11,22 X4,19,1,20
Gauss code {1, -4, 3, -11}, {-5, -1, 6, -3, 9, -10, -2, 7}, {8, 5, -7, 2, 4, -6, 11, -8, 10, -9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n335 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) (w-1) (v w-2 v-2 w+1)}{\sqrt{u} v w} (db)
Jones polynomial -q^2+5 q-9+13 q^{-1} -16 q^{-2} +17 q^{-3} -14 q^{-4} +12 q^{-5} -6 q^{-6} +3 q^{-7} (db)
Signature -2 (db)
HOMFLY-PT polynomial a^8 z^{-2} +2 z^2 a^6-2 a^6 z^{-2} -a^6-3 z^4 a^4-4 z^2 a^4+a^4 z^{-2} +z^6 a^2+2 z^4 a^2+2 z^2 a^2-z^4+1 (db)
Kauffman polynomial 2 a^5 z^9+2 a^3 z^9+5 a^6 z^8+12 a^4 z^8+7 a^2 z^8+3 a^7 z^7+9 a^5 z^7+15 a^3 z^7+9 a z^7-9 a^6 z^6-18 a^4 z^6-4 a^2 z^6+5 z^6-3 a^7 z^5-23 a^5 z^5-36 a^3 z^5-15 a z^5+z^5 a^{-1} +6 a^8 z^4+16 a^6 z^4+7 a^4 z^4-9 a^2 z^4-6 z^4+7 a^7 z^3+21 a^5 z^3+19 a^3 z^3+5 a z^3-11 a^8 z^2-16 a^6 z^2-3 a^4 z^2+2 a^2 z^2-9 a^7 z-9 a^5 z+6 a^8+9 a^6+3 a^4+1+2 a^7 z^{-1} +2 a^5 z^{-1} -a^8 z^{-2} -2 a^6 z^{-2} -a^4 z^{-2} (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 \
-6-5-4-3-2-10123χ
5         1-1
3        4 4
1       51 -4
-1      84  4
-3     107   -3
-5    76    1
-7   710     3
-9  57      -2
-11 17       6
-1325        -3
-153         3
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-6 {\mathbb Z}^{3} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\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.

L11n334.gif

L11n334

L11n336.gif

L11n336