L11a364

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L11a363.gif

L11a363

L11a365.gif

L11a365

Contents

L11a364.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a364 at Knotilus!


Link Presentations

[edit Notes on L11a364's Link Presentations]

Planar diagram presentation X12,1,13,2 X2,13,3,14 X14,3,15,4 X4,11,5,12 X16,6,17,5 X22,16,11,15 X18,8,19,7 X20,10,21,9 X6,18,7,17 X8,20,9,19 X10,22,1,21
Gauss code {1, -2, 3, -4, 5, -9, 7, -10, 8, -11}, {4, -1, 2, -3, 6, -5, 9, -7, 10, -8, 11, -6}
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation L11a364 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(2)^4 t(1)^4-t(2)^3 t(1)^4-t(2)^4 t(1)^3+3 t(2)^3 t(1)^3-2 t(2)^2 t(1)^3-2 t(2)^3 t(1)^2+3 t(2)^2 t(1)^2-2 t(2) t(1)^2-2 t(2)^2 t(1)+3 t(2) t(1)-t(1)-t(2)+1}{t(1)^2 t(2)^2} (db)
Jones polynomial q^{17/2}-2 q^{15/2}+3 q^{13/2}-5 q^{11/2}+6 q^{9/2}-7 q^{7/2}+6 q^{5/2}-6 q^{3/2}+4 \sqrt{q}-\frac{3}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{1}{q^{5/2}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^7 a^{-5} +6 z^5 a^{-5} +11 z^3 a^{-5} +6 z a^{-5} -z^9 a^{-3} -8 z^7 a^{-3} -23 z^5 a^{-3} -28 z^3 a^{-3} -12 z a^{-3} - a^{-3} z^{-1} +z^7 a^{-1} +6 z^5 a^{-1} +11 z^3 a^{-1} +7 z a^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial -z^{10} a^{-2} -z^{10} a^{-4} -2 z^9 a^{-1} -4 z^9 a^{-3} -2 z^9 a^{-5} +3 z^8 a^{-2} +3 z^8 a^{-4} -2 z^8 a^{-6} -2 z^8-a z^7+10 z^7 a^{-1} +22 z^7 a^{-3} +9 z^7 a^{-5} -2 z^7 a^{-7} +z^6 a^{-2} -z^6 a^{-4} +6 z^6 a^{-6} -2 z^6 a^{-8} +10 z^6+5 a z^5-16 z^5 a^{-1} -45 z^5 a^{-3} -18 z^5 a^{-5} +4 z^5 a^{-7} -2 z^5 a^{-9} -6 z^4 a^{-2} -5 z^4 a^{-4} -8 z^4 a^{-6} +3 z^4 a^{-8} -z^4 a^{-10} -13 z^4-6 a z^3+14 z^3 a^{-1} +43 z^3 a^{-3} +15 z^3 a^{-5} -4 z^3 a^{-7} +4 z^3 a^{-9} +5 z^2 a^{-2} +5 z^2 a^{-4} +z^2 a^{-6} -z^2 a^{-8} +2 z^2 a^{-10} +4 z^2+a z-7 z a^{-1} -14 z a^{-3} -5 z a^{-5} -z a^{-9} - a^{-2} + a^{-1} z^{-1} + a^{-3} 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 \
-4-3-2-101234567χ
18           1-1
16          1 1
14         21 -1
12        31  2
10       32   -1
8      43    1
6     34     1
4    33      0
2   24       2
0  12        -1
-2 12         1
-4 1          -1
-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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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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L11a363.gif

L11a363

L11a365.gif

L11a365