L11a399

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L11a398.gif

L11a398

L11a400.gif

L11a400

Contents

L11a399.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a399's Link Presentations]

Planar diagram presentation X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X4,15,1,16 X18,22,19,21 X20,9,21,10 X10,19,5,20 X16,12,17,11 X22,18,11,17
Gauss code {1, -4, 3, -6}, {2, -1, 5, -3, 8, -9}, {10, -2, 4, -5, 6, -10, 11, -7, 9, -8, 7, -11}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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A Morse Link Presentation L11a399 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 (t(1)-1) (t(2)-1)^2 (t(3)-1)^2}{\sqrt{t(1)} t(2) t(3)} (db)
Jones polynomial q^6-4 q^5- q^{-5} +8 q^4+4 q^{-4} -13 q^3-7 q^{-3} +18 q^2+14 q^{-2} -20 q-17 q^{-1} +21 (db)
Signature 0 (db)
HOMFLY-PT polynomial z^4 a^{-4} -a^4 z^2+z^2 a^{-4} +a^4 z^{-2} -z^6 a^{-2} +2 a^2 z^4-2 z^4 a^{-2} +2 a^2 z^2-2 z^2 a^{-2} -2 a^2 z^{-2} -a^2-z^6-z^4+ z^{-2} +1 (db)
Kauffman polynomial z^6 a^{-6} -2 z^4 a^{-6} +4 z^7 a^{-5} +a^5 z^5-10 z^5 a^{-5} -a^5 z^3+5 z^3 a^{-5} +7 z^8 a^{-4} +4 a^4 z^6-19 z^6 a^{-4} -7 a^4 z^4+15 z^4 a^{-4} +4 a^4 z^2-4 z^2 a^{-4} -a^4 z^{-2} +a^4+6 z^9 a^{-3} +6 a^3 z^7-11 z^7 a^{-3} -7 a^3 z^5+2 z^5 a^{-3} +a^3 z^3+3 z^3 a^{-3} -a^3 z+2 a^3 z^{-1} +2 z^{10} a^{-2} +6 a^2 z^8+9 z^8 a^{-2} -a^2 z^6-32 z^6 a^{-2} -12 a^2 z^4+30 z^4 a^{-2} +8 a^2 z^2-8 z^2 a^{-2} -2 a^2 z^{-2} +a^2+5 a z^9+11 z^9 a^{-1} -a z^7-22 z^7 a^{-1} -3 a z^5+17 z^5 a^{-1} -2 a z^3-6 z^3 a^{-1} -a z+2 a z^{-1} +2 z^{10}+8 z^8-17 z^6+8 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 \
-5-4-3-2-10123456χ
13           11
11          3 -3
9         51 4
7        83  -5
5       105   5
3      108    -2
1     1110     1
-1    812      4
-3   69       -3
-5  310        7
-7 14         -3
-9 3          3
-111           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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

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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L11a398

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