L10a170

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L10a169.gif

L10a169

L10a171.gif

L10a171

Contents

L10a170.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a170 at Knotilus!


Link Presentations

[edit Notes on L10a170's Link Presentations]

Planar diagram presentation X6172 X12,6,13,5 X8493 X2,16,3,15 X16,7,17,8 X14,19,11,20 X20,13,15,14 X18,10,19,9 X10,12,5,11 X4,17,1,18
Gauss code {1, -4, 3, -10}, {9, -2, 7, -6}, {2, -1, 5, -3, 8, -9}, {4, -5, 10, -8, 6, -7}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L10a170 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (w-1) (x-1) (v w x+v (-w)+v-w x+x-1)}{\sqrt{u} \sqrt{v} w x} (db)
Jones polynomial -4 q^{9/2}+\frac{1}{q^{9/2}}+8 q^{7/2}-\frac{4}{q^{7/2}}-13 q^{5/2}+\frac{7}{q^{5/2}}+14 q^{3/2}-\frac{13}{q^{3/2}}+q^{11/2}-18 \sqrt{q}+\frac{13}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z^5 a^{-3} -a^3 z^3+2 z^3 a^{-3} +a^3 z^{-3} - a^{-3} z^{-3} -a^3 z+z a^{-3} +a^3 z^{-1} - a^{-3} z^{-1} -z^7 a^{-1} +2 a z^5-4 z^5 a^{-1} +5 a z^3-6 z^3 a^{-1} -3 a z^{-3} +3 a^{-1} z^{-3} +2 a z-2 z a^{-1} -3 a z^{-1} +3 a^{-1} z^{-1} (db)
Kauffman polynomial z^4 a^{-6} +4 z^5 a^{-5} -2 z^3 a^{-5} +a^4 z^6+8 z^6 a^{-4} -2 a^4 z^4-7 z^4 a^{-4} +a^4 z^2+z^2 a^{-4} +4 a^3 z^7+11 z^7 a^{-3} -11 a^3 z^5-17 z^5 a^{-3} +10 a^3 z^3+12 z^3 a^{-3} +a^3 z^{-3} + a^{-3} z^{-3} -3 a^3 z-3 z a^{-3} -2 a^3 z^{-1} -2 a^{-3} z^{-1} +5 a^2 z^8+8 z^8 a^{-2} -9 a^2 z^6-6 z^6 a^{-2} -a^2 z^4-6 z^4 a^{-2} +3 a^2 z^2+3 z^2 a^{-2} -3 a^2 z^{-2} -3 a^{-2} z^{-2} +4 a^2+4 a^{-2} +2 a z^9+2 z^9 a^{-1} +11 a z^7+18 z^7 a^{-1} -43 a z^5-53 z^5 a^{-1} +40 a z^3+44 z^3 a^{-1} +3 a z^{-3} +3 a^{-1} z^{-3} -11 a z-11 z a^{-1} -3 a z^{-1} -3 a^{-1} z^{-1} +13 z^8-24 z^6+3 z^4+4 z^2-6 z^{-2} +7 (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 \
-5-4-3-2-1012345χ
12          1-1
10         3 3
8        51 -4
6       83  5
4      87   -1
2     106    4
0    712     5
-2   66      0
-4  39       6
-6 14        -3
-8 3         3
-101          -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
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}^{9}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\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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L10a169.gif

L10a169

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L10a171