L10a113

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L10a112.gif

L10a112

L10a114.gif

L10a114

Contents

L10a113.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a113's Link Presentations]

Planar diagram presentation X12,1,13,2 X8493 X18,14,19,13 X20,8,11,7 X6,20,7,19 X4,15,5,16 X14,5,15,6 X16,10,17,9 X2,11,3,12 X10,18,1,17
Gauss code {1, -9, 2, -6, 7, -5, 4, -2, 8, -10}, {9, -1, 3, -7, 6, -8, 10, -3, 5, -4}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation L10a113 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) \left(t(1)^2-t(1)+1\right) (t(2)-1) \left(t(2)^2-t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial 10 q^{9/2}-11 q^{7/2}+11 q^{5/2}-\frac{1}{q^{5/2}}-12 q^{3/2}+\frac{3}{q^{3/2}}-q^{15/2}+3 q^{13/2}-6 q^{11/2}+8 \sqrt{q}-\frac{6}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^7 a^{-3} -2 z^5 a^{-1} +5 z^5 a^{-3} -z^5 a^{-5} +a z^3-7 z^3 a^{-1} +10 z^3 a^{-3} -3 z^3 a^{-5} +2 a z-7 z a^{-1} +8 z a^{-3} -3 z a^{-5} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -2 z^9 a^{-1} -2 z^9 a^{-3} -10 z^8 a^{-2} -7 z^8 a^{-4} -3 z^8-a z^7+z^7 a^{-1} -9 z^7 a^{-3} -11 z^7 a^{-5} +32 z^6 a^{-2} +10 z^6 a^{-4} -10 z^6 a^{-6} +12 z^6+4 a z^5+18 z^5 a^{-1} +44 z^5 a^{-3} +24 z^5 a^{-5} -6 z^5 a^{-7} -23 z^4 a^{-2} +10 z^4 a^{-4} +16 z^4 a^{-6} -3 z^4 a^{-8} -14 z^4-6 a z^3-30 z^3 a^{-1} -44 z^3 a^{-3} -16 z^3 a^{-5} +3 z^3 a^{-7} -z^3 a^{-9} -11 z^2 a^{-4} -8 z^2 a^{-6} +3 z^2+4 a z+14 z a^{-1} +16 z a^{-3} +6 z a^{-5} +1-a z^{-1} - a^{-1} 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 \
-4-3-2-10123456χ
16          11
14         2 -2
12        41 3
10       62  -4
8      54   1
6     66    0
4    65     1
2   48      4
0  24       -2
-2 14        3
-4 2         -2
-61          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=2 i=4
r=-4 {\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}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\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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L10a112.gif

L10a112

L10a114.gif

L10a114