L10a162

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L10a161.gif

L10a161

L10a163.gif

L10a163

Contents

L10a162.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a162's Link Presentations]

Planar diagram presentation X8192 X16,6,17,5 X10,20,11,19 X18,10,19,9 X20,12,13,11 X2,13,3,14 X14,3,15,4 X4758 X12,16,7,15 X6,18,1,17
Gauss code {1, -6, 7, -8, 2, -10}, {8, -1, 4, -3, 5, -9}, {6, -7, 9, -2, 10, -4, 3, -5}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation L10a162 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(2)-1) (t(3) t(1)-t(1)+1) (t(1) t(3)-t(3)+1) (t(2) t(3)+1)}{t(1) t(2) t(3)^{3/2}} (db)
Jones polynomial q^7-3 q^6+6 q^5-9 q^4+12 q^3-11 q^2+12 q-8+6 q^{-1} -3 q^{-2} + q^{-3} (db)
Signature 2 (db)
HOMFLY-PT polynomial z^6 a^{-4} +4 z^4 a^{-4} +5 z^2 a^{-4} + a^{-4} z^{-2} +2 a^{-4} -z^8 a^{-2} -6 z^6 a^{-2} -13 z^4 a^{-2} -12 z^2 a^{-2} -2 a^{-2} z^{-2} -5 a^{-2} +z^6+4 z^4+5 z^2+ z^{-2} +3 (db)
Kauffman polynomial z^4 a^{-8} -z^2 a^{-8} +3 z^5 a^{-7} -3 z^3 a^{-7} +5 z^6 a^{-6} -6 z^4 a^{-6} +3 z^2 a^{-6} - a^{-6} +6 z^7 a^{-5} -9 z^5 a^{-5} +6 z^3 a^{-5} +5 z^8 a^{-4} -8 z^6 a^{-4} +7 z^4 a^{-4} -3 z^2 a^{-4} - a^{-4} z^{-2} +2 a^{-4} +2 z^9 a^{-3} +3 z^7 a^{-3} -13 z^5 a^{-3} +12 z^3 a^{-3} -5 z a^{-3} +2 a^{-3} z^{-1} +9 z^8 a^{-2} +a^2 z^6-26 z^6 a^{-2} -3 a^2 z^4+29 z^4 a^{-2} +2 a^2 z^2-18 z^2 a^{-2} -2 a^{-2} z^{-2} +6 a^{-2} +2 z^9 a^{-1} +3 a z^7-9 a z^5-10 z^5 a^{-1} +6 a z^3+9 z^3 a^{-1} -5 z a^{-1} +2 a^{-1} z^{-1} +4 z^8-12 z^6+12 z^4-9 z^2- z^{-2} +4 (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
15          11
13         2 -2
11        41 3
9       63  -3
7      63   3
5     67    1
3    65     1
1   37      4
-1  35       -2
-3 14        3
-5 2         -2
-71          1
Integral Khovanov Homology

(db, data source)

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

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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