L11n50

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L11n49.gif

L11n49

L11n51.gif

L11n51

Contents

L11n50.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n50's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X12,8,13,7 X13,18,14,19 X9,17,10,16 X17,9,18,8 X19,22,20,5 X15,21,16,20 X21,15,22,14 X2536 X4,12,1,11
Gauss code {1, -10, 2, -11}, {10, -1, 3, 6, -5, -2, 11, -3, -4, 9, -8, 5, -6, 4, -7, 8, -9, 7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n50 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(t(2)^2-4 t(2)+1\right)}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial -7 q^{9/2}+8 q^{7/2}-8 q^{5/2}+7 q^{3/2}-\frac{1}{q^{3/2}}+q^{15/2}-3 q^{13/2}+5 q^{11/2}-6 \sqrt{q}+\frac{2}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z a^{-7} -2 z^3 a^{-5} -2 z a^{-5} +z^5 a^{-3} +2 z^3 a^{-3} +2 z a^{-3} -2 z^3 a^{-1} +a z-2 z a^{-1} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial z^6 a^{-8} -3 z^4 a^{-8} +2 z^2 a^{-8} +3 z^7 a^{-7} -10 z^5 a^{-7} +8 z^3 a^{-7} -2 z a^{-7} +3 z^8 a^{-6} -8 z^6 a^{-6} +3 z^4 a^{-6} -z^2 a^{-6} +z^9 a^{-5} +3 z^7 a^{-5} -17 z^5 a^{-5} +15 z^3 a^{-5} -4 z a^{-5} +5 z^8 a^{-4} -15 z^6 a^{-4} +14 z^4 a^{-4} -5 z^2 a^{-4} +z^9 a^{-3} +z^7 a^{-3} -9 z^5 a^{-3} +13 z^3 a^{-3} -4 z a^{-3} +2 z^8 a^{-2} -6 z^6 a^{-2} +10 z^4 a^{-2} -3 z^2 a^{-2} +z^7 a^{-1} -2 z^5 a^{-1} +a z^3+7 z^3 a^{-1} -2 a z-4 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +2 z^4-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 \
-2-101234567χ
16         1-1
14        2 2
12       31 -2
10      42  2
8     43   -1
6    44    0
4   34     1
2  34      -1
0 15       4
-2 1        -1
-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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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).

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L11n49.gif

L11n49

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L11n51