L11a298

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L11a297.gif

L11a297

L11a299.gif

L11a299

Contents

L11a298.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a298's Link Presentations]

Planar diagram presentation X10,1,11,2 X12,3,13,4 X18,8,19,7 X16,6,17,5 X22,13,9,14 X20,15,21,16 X6,18,7,17 X14,21,15,22 X4,20,5,19 X2,9,3,10 X8,11,1,12
Gauss code {1, -10, 2, -9, 4, -7, 3, -11}, {10, -1, 11, -2, 5, -8, 6, -4, 7, -3, 9, -6, 8, -5}
A Braid Representative
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A Morse Link Presentation L11a298 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(2 t(2) t(1)^2+2 t(2)^2 t(1)-t(2) t(1)+2 t(1)+2 t(2)\right)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -q^{9/2}+3 q^{7/2}-6 q^{5/2}+8 q^{3/2}-10 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{11}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{2}{q^{11/2}}+\frac{1}{q^{13/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^3 a^5-2 z a^5+z^5 a^3+2 z^3 a^3+2 z^5 a+6 z^3 a+5 z a+a z^{-1} +z^5 a^{-1} +z^3 a^{-1} -2 z a^{-1} - a^{-1} z^{-1} -z^3 a^{-3} -z a^{-3} (db)
Kauffman polynomial -a^4 z^{10}-a^2 z^{10}-2 a^5 z^9-5 a^3 z^9-3 a z^9-a^6 z^8+a^4 z^8-4 a^2 z^8-6 z^8+11 a^5 z^7+19 a^3 z^7-8 z^7 a^{-1} +6 a^6 z^6+11 a^4 z^6+23 a^2 z^6-8 z^6 a^{-2} +10 z^6-20 a^5 z^5-17 a^3 z^5+23 a z^5+14 z^5 a^{-1} -6 z^5 a^{-3} -12 a^6 z^4-22 a^4 z^4-22 a^2 z^4+11 z^4 a^{-2} -3 z^4 a^{-4} +2 z^4+14 a^5 z^3-26 a z^3-6 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} +8 a^6 z^2+9 a^4 z^2+4 a^2 z^2-4 z^2 a^{-2} -z^2-4 a^5 z+10 a z+4 z a^{-1} -2 z a^{-3} +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 \
-7-6-5-4-3-2-101234χ
10           11
8          2 -2
6         41 3
4        42  -2
2       64   2
0      76    -1
-2     44     0
-4    47      3
-6   34       -1
-8  14        3
-10 13         -2
-12 1          1
-141           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\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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L11a297

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L11a299