L11n314

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L11n313.gif

L11n313

L11n315.gif

L11n315

Contents

L11n314.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n314's Link Presentations]

Planar diagram presentation X6172 X3,12,4,13 X7,17,8,16 X9,20,10,21 X11,18,12,19 X19,22,20,11 X15,9,16,8 X21,10,22,5 X17,14,18,15 X2536 X13,4,14,1
Gauss code {1, -10, -2, 11}, {10, -1, -3, 7, -4, 8}, {-5, 2, -11, 9, -7, 3, -9, 5, -6, 4, -8, 6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n314 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(1) t(3)^2 t(2)^2+t(1) t(2)^2-3 t(1) t(3) t(2)^2+t(3) t(2)^2-t(2)^2-3 t(1) t(3)^2 t(2)+t(3)^2 t(2)-t(1) t(2)+3 t(1) t(3) t(2)-3 t(3) t(2)+3 t(2)+t(1) t(3)^2-t(3)^2-t(1) t(3)+3 t(3)-2}{\sqrt{t(1)} t(2) t(3)} (db)
Jones polynomial 1-2 q^{-1} +6 q^{-2} -7 q^{-3} +10 q^{-4} -10 q^{-5} +10 q^{-6} -7 q^{-7} +5 q^{-8} -2 q^{-9} (db)
Signature -4 (db)
HOMFLY-PT polynomial -a^{10}+a^8 z^4+3 a^8 z^2+2 a^8-a^6 z^6-3 a^6 z^4-2 a^6 z^2+a^6 z^{-2} -a^4 z^6-3 a^4 z^4-3 a^4 z^2-2 a^4 z^{-2} -4 a^4+a^2 z^4+3 a^2 z^2+a^2 z^{-2} +3 a^2 (db)
Kauffman polynomial 3 a^{11} z^3-2 a^{11} z+a^{10} z^6+4 a^{10} z^4-5 a^{10} z^2+2 a^{10}+3 a^9 z^7-4 a^9 z^5+8 a^9 z^3-4 a^9 z+3 a^8 z^8-5 a^8 z^6+11 a^8 z^4-12 a^8 z^2+4 a^8+a^7 z^9+4 a^7 z^7-9 a^7 z^5+4 a^7 z^3+5 a^6 z^8-8 a^6 z^6+a^6 z^2+a^6 z^{-2} -2 a^6+a^5 z^9+3 a^5 z^7-10 a^5 z^5+6 a^5 z-2 a^5 z^{-1} +2 a^4 z^8-a^4 z^6-11 a^4 z^4+14 a^4 z^2+2 a^4 z^{-2} -7 a^4+2 a^3 z^7-5 a^3 z^5+a^3 z^3+4 a^3 z-2 a^3 z^{-1} +a^2 z^6-4 a^2 z^4+6 a^2 z^2+a^2 z^{-2} -4 a^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 \
-7-6-5-4-3-2-1012χ
1         11
-1        1 -1
-3       51 4
-5      32  -1
-7     74   3
-9    55    0
-11   55     0
-13  36      3
-15 24       -2
-17 3        3
-192         -2
Integral Khovanov Homology

(db, data source)

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

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See/edit the Link Page master template (intermediate).

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L11n313

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