L10a110

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L10a109.gif

L10a109

L10a111.gif

L10a111

Contents

L10a110.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a110 at Knotilus!


Link Presentations

[edit Notes on L10a110's Link Presentations]

Planar diagram presentation X12,1,13,2 X8493 X18,14,19,13 X20,16,11,15 X14,20,15,19 X6,17,7,18 X16,7,17,8 X10,6,1,5 X4,10,5,9 X2,11,3,12
Gauss code {1, -10, 2, -9, 8, -6, 7, -2, 9, -8}, {10, -1, 3, -5, 4, -7, 6, -3, 5, -4}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L10a110 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(2)^3 t(1)^3-2 t(2)^2 t(1)^3+2 t(2) t(1)^3-2 t(1)^3-2 t(2)^3 t(1)^2+2 t(2)^2 t(1)^2-2 t(2) t(1)^2+2 t(1)^2+2 t(2)^3 t(1)-2 t(2)^2 t(1)+2 t(2) t(1)-2 t(1)-2 t(2)^3+2 t(2)^2-2 t(2)+1}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -q^{15/2}+3 q^{13/2}-6 q^{11/2}+9 q^{9/2}-9 q^{7/2}+10 q^{5/2}-9 q^{3/2}+6 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}} (db)
Signature 3 (db)
HOMFLY-PT polynomial -z^5 a^{-5} -3 z^3 a^{-5} -3 z a^{-5} - a^{-5} z^{-1} +z^7 a^{-3} +5 z^5 a^{-3} +10 z^3 a^{-3} +11 z a^{-3} +5 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-9 z^3 a^{-1} +4 a z-14 z a^{-1} +4 a z^{-1} -8 a^{-1} z^{-1} (db)
Kauffman polynomial -z^9 a^{-1} -z^9 a^{-3} -5 z^8 a^{-2} -4 z^8 a^{-4} -z^8-a z^7-7 z^7 a^{-3} -8 z^7 a^{-5} +11 z^6 a^{-2} -9 z^6 a^{-6} +2 z^6+6 a z^5+12 z^5 a^{-1} +26 z^5 a^{-3} +14 z^5 a^{-5} -6 z^5 a^{-7} +9 z^4 a^{-2} +22 z^4 a^{-4} +15 z^4 a^{-6} -3 z^4 a^{-8} +5 z^4-13 a z^3-24 z^3 a^{-1} -20 z^3 a^{-3} -4 z^3 a^{-5} +4 z^3 a^{-7} -z^3 a^{-9} -28 z^2 a^{-2} -23 z^2 a^{-4} -9 z^2 a^{-6} -14 z^2+12 a z+20 z a^{-1} +9 z a^{-3} -z a^{-7} +14 a^{-2} +9 a^{-4} +2 a^{-6} +8-4 a z^{-1} -8 a^{-1} z^{-1} -5 a^{-3} z^{-1} - a^{-5} 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       52  -3
8      44   0
6     65    -1
4    34     -1
2   47      3
0  12       -1
-2  4        4
-411         0
-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} {\mathbb Z}
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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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L10a109.gif

L10a109

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L10a111