L11n327

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L11n326.gif

L11n326

L11n328.gif

L11n328

Contents

L11n327.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n327 at Knotilus!


Link Presentations

[edit Notes on L11n327's Link Presentations]

Planar diagram presentation X6172 X5,14,6,15 X3849 X2,16,3,15 X16,7,17,8 X13,18,14,19 X9,21,10,20 X19,5,20,12 X11,13,12,22 X21,11,22,10 X17,1,18,4
Gauss code {1, -4, -3, 11}, {-2, -1, 5, 3, -7, 10, -9, 8}, {-6, 2, 4, -5, -11, 6, -8, 7, -10, 9}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
A Morse Link Presentation L11n327 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v^2 w^3-2 u v^2 w^2-u v w^3+3 u v w^2-2 u v w-u w^2+u w+v^3 \left(-w^2\right)+v^3 w+2 v^2 w^2-3 v^2 w+v^2+2 v w-v}{\sqrt{u} v^{3/2} w^{3/2}} (db)
Jones polynomial 2 q^5-4 q^4+6 q^3-6 q^2+8 q-6+6 q^{-1} -3 q^{-2} +2 q^{-3} - q^{-4} (db)
Signature 2 (db)
HOMFLY-PT polynomial 2 z^2 a^{-4} + a^{-4} z^{-2} +3 a^{-4} -a^2 z^4-3 z^4 a^{-2} -3 a^2 z^2-10 z^2 a^{-2} -2 a^{-2} z^{-2} -2 a^2-9 a^{-2} +z^6+5 z^4+10 z^2+ z^{-2} +8 (db)
Kauffman polynomial 3 z^2 a^{-6} - a^{-6} +z^5 a^{-5} +3 z^3 a^{-5} -z a^{-5} +3 z^6 a^{-4} -3 z^4 a^{-4} - a^{-4} z^{-2} +3 a^{-4} +a^3 z^7+4 z^7 a^{-3} -5 a^3 z^5-10 z^5 a^{-3} +7 a^3 z^3+12 z^3 a^{-3} -2 a^3 z-8 z a^{-3} +2 a^{-3} z^{-1} +2 a^2 z^8+3 z^8 a^{-2} -10 a^2 z^6-8 z^6 a^{-2} +16 a^2 z^4+12 z^4 a^{-2} -10 a^2 z^2-19 z^2 a^{-2} -2 a^{-2} z^{-2} +3 a^2+11 a^{-2} +a z^9+z^9 a^{-1} -a z^7+2 z^7 a^{-1} -9 a z^5-15 z^5 a^{-1} +16 a z^3+18 z^3 a^{-1} -7 a z-12 z a^{-1} +2 a^{-1} z^{-1} +5 z^8-21 z^6+31 z^4-26 z^2- z^{-2} +11 (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 \
-5-4-3-2-101234χ
11         22
9        31-2
7       31 2
5      33  0
3     53   2
1    46    2
-1   22     0
-3  14      3
-5 12       -1
-7 1        1
-91         -1
Integral Khovanov Homology

(db, data source)

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

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L11n326.gif

L11n326

L11n328.gif

L11n328