L11n447

From Knot Atlas
Jump to: navigation, search

L11n446.gif

L11n446

L11n448.gif

L11n448

Contents

L11n447.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n447 at Knotilus!


Link Presentations

[edit Notes on L11n447's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,18,12,19 X19,17,20,22 X21,9,22,16 X15,21,16,20 X17,12,18,13 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, 3, -4}, {-9, 5, -6, 8, -7, 6}, {11, -2, -5, 9, 4, -3, -8, 7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n447 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(w-1) (x-1) \left(-u v+u x^2-u x+u+v x^2-v x+v-x^2\right)}{\sqrt{u} \sqrt{v} \sqrt{w} x^{3/2}} (db)
Jones polynomial -\frac{12}{q^{9/2}}+\frac{9}{q^{7/2}}-\frac{11}{q^{5/2}}+\frac{8}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{6}{q^{13/2}}+\frac{7}{q^{11/2}}+2 \sqrt{q}-\frac{6}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^9 z^{-1} +a^9 z^{-3} -4 z a^7-7 a^7 z^{-1} -3 a^7 z^{-3} +5 z^3 a^5+13 z a^5+12 a^5 z^{-1} +3 a^5 z^{-3} -2 z^5 a^3-7 z^3 a^3-11 z a^3-7 a^3 z^{-1} -a^3 z^{-3} +2 z^3 a+2 z a+a z^{-1} (db)
Kauffman polynomial a^9 z^7-5 a^9 z^5+10 a^9 z^3-a^9 z^{-3} -10 a^9 z+5 a^9 z^{-1} +2 a^8 z^8-7 a^8 z^6+5 a^8 z^4+6 a^8 z^2+3 a^8 z^{-2} -9 a^8+a^7 z^9+5 a^7 z^7-34 a^7 z^5+55 a^7 z^3-3 a^7 z^{-3} -38 a^7 z+14 a^7 z^{-1} +8 a^6 z^8-22 a^6 z^6+a^6 z^4+27 a^6 z^2+6 a^6 z^{-2} -21 a^6+a^5 z^9+15 a^5 z^7-65 a^5 z^5+83 a^5 z^3-3 a^5 z^{-3} -54 a^5 z+18 a^5 z^{-1} +6 a^4 z^8-8 a^4 z^6-18 a^4 z^4+33 a^4 z^2+3 a^4 z^{-2} -18 a^4+11 a^3 z^7-35 a^3 z^5+45 a^3 z^3-a^3 z^{-3} -31 a^3 z+11 a^3 z^{-1} +7 a^2 z^6-14 a^2 z^4+15 a^2 z^2-6 a^2+a z^5+7 a z^3-5 a z+2 a z^{-1} +3 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 \
-8-7-6-5-4-3-2-101χ
2         2-2
0        4 4
-2       64 -2
-4      531 3
-6     46   2
-8    85    3
-10   510     5
-12  12      -1
-14 15       4
-16 1        -1
-181         1
Integral Khovanov Homology

(db, data source)

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

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.

L11n446.gif

L11n446

L11n448.gif

L11n448