L11n37

From Knot Atlas
Jump to: navigation, search

L11n36.gif

L11n36

L11n38.gif

L11n38

Contents

L11n37.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n37 at Knotilus!


Link Presentations

[edit Notes on L11n37's Link Presentations]

Planar diagram presentation X6172 X20,7,21,8 X4,21,1,22 X11,14,12,15 X8493 X5,13,6,12 X13,5,14,22 X15,19,16,18 X9,17,10,16 X17,11,18,10 X2,20,3,19
Gauss code {1, -11, 5, -3}, {-6, -1, 2, -5, -9, 10, -4, 6, -7, 4, -8, 9, -10, 8, 11, -2, 3, 7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gif
A Morse Link Presentation L11n37 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(2 v^2-3 v+2\right)}{\sqrt{u} v^{3/2}} (db)
Jones polynomial q^{15/2}-3 q^{13/2}+6 q^{11/2}-8 q^{9/2}+9 q^{7/2}-10 q^{5/2}+8 q^{3/2}-7 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{1}{q^{3/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial 2 z^5 a^{-3} -3 z^3 a^{-1} +7 z^3 a^{-3} -3 z^3 a^{-5} +a z-5 z a^{-1} +9 z a^{-3} -6 z a^{-5} +z a^{-7} +a z^{-1} -2 a^{-1} z^{-1} +3 a^{-3} z^{-1} -3 a^{-5} z^{-1} + a^{-7} z^{-1} (db)
Kauffman polynomial -z^9 a^{-3} -z^9 a^{-5} -3 z^8 a^{-2} -6 z^8 a^{-4} -3 z^8 a^{-6} -2 z^7 a^{-1} -5 z^7 a^{-3} -6 z^7 a^{-5} -3 z^7 a^{-7} +9 z^6 a^{-2} +15 z^6 a^{-4} +5 z^6 a^{-6} -z^6 a^{-8} +6 z^5 a^{-1} +24 z^5 a^{-3} +27 z^5 a^{-5} +9 z^5 a^{-7} -15 z^4 a^{-2} -9 z^4 a^{-4} +6 z^4 a^{-6} +3 z^4 a^{-8} -3 z^4-a z^3-16 z^3 a^{-1} -36 z^3 a^{-3} -28 z^3 a^{-5} -7 z^3 a^{-7} +7 z^2 a^{-2} +z^2 a^{-4} -7 z^2 a^{-6} -3 z^2 a^{-8} +2 z^2+2 a z+11 z a^{-1} +20 z a^{-3} +14 z a^{-5} +3 z a^{-7} -2 a^{-2} +2 a^{-6} + a^{-8} -a z^{-1} -2 a^{-1} z^{-1} -3 a^{-3} z^{-1} -3 a^{-5} z^{-1} - a^{-7} z^{-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 \
-2-101234567χ
16         1-1
14        2 2
12       41 -3
10      42  2
8     54   -1
6    54    1
4   35     2
2  45      -1
0 15       4
-2 2        -2
-41         1
Integral Khovanov Homology

(db, data source)

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

L11n36.gif

L11n36

L11n38.gif

L11n38