L11a171

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L11a170.gif

L11a170

L11a172.gif

L11a172

Contents

L11a171.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a171 at Knotilus!


Link Presentations

[edit Notes on L11a171's Link Presentations]

Planar diagram presentation X8192 X20,9,21,10 X14,5,15,6 X18,12,19,11 X10,4,11,3 X12,7,13,8 X16,13,17,14 X22,17,7,18 X6,15,1,16 X4,21,5,22 X2,20,3,19
Gauss code {1, -11, 5, -10, 3, -9}, {6, -1, 2, -5, 4, -6, 7, -3, 9, -7, 8, -4, 11, -2, 10, -8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a171 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(1)^2 t(2)^4-3 t(1) t(2)^4+t(2)^4-7 t(1)^2 t(2)^3+12 t(1) t(2)^3-5 t(2)^3+9 t(1)^2 t(2)^2-17 t(1) t(2)^2+9 t(2)^2-5 t(1)^2 t(2)+12 t(1) t(2)-7 t(2)+t(1)^2-3 t(1)+2}{t(1) t(2)^2} (db)
Jones polynomial q^{3/2}-5 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{20}{q^{3/2}}+\frac{26}{q^{5/2}}-\frac{31}{q^{7/2}}+\frac{31}{q^{9/2}}-\frac{27}{q^{11/2}}+\frac{20}{q^{13/2}}-\frac{12}{q^{15/2}}+\frac{5}{q^{17/2}}-\frac{1}{q^{19/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^7 z^5+a^7 z^3-a^5 z^7-2 a^5 z^5-a^5 z^3-a^3 z^7-2 a^3 z^5-a^3 z^3+a^3 z^{-1} +a z^5+a z^3-a z-a z^{-1} (db)
Kauffman polynomial a^{11} z^5+5 a^{10} z^6-3 a^{10} z^4+12 a^9 z^7-14 a^9 z^5+5 a^9 z^3+17 a^8 z^8-25 a^8 z^6+13 a^8 z^4-2 a^8 z^2+13 a^7 z^9-7 a^7 z^7-14 a^7 z^5+8 a^7 z^3+4 a^6 z^{10}+26 a^6 z^8-64 a^6 z^6+38 a^6 z^4-6 a^6 z^2+23 a^5 z^9-34 a^5 z^7+a^5 z^5+6 a^5 z^3+4 a^4 z^{10}+19 a^4 z^8-54 a^4 z^6+34 a^4 z^4-6 a^4 z^2+10 a^3 z^9-10 a^3 z^7-9 a^3 z^5+8 a^3 z^3+a^3 z-a^3 z^{-1} +10 a^2 z^8-19 a^2 z^6+11 a^2 z^4-2 a^2 z^2+a^2+5 a z^7-9 a z^5+5 a z^3+a z-a z^{-1} +z^6-z^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 \
-8-7-6-5-4-3-2-10123χ
4           1-1
2          4 4
0         71 -6
-2        134  9
-4       148   -6
-6      1712    5
-8     1515     0
-10    1216      -4
-12   815       7
-14  412        -8
-16 18         7
-18 4          -4
-201           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-4 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=-3 {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{15}
r=-2 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{16} {\mathbb Z}^{17}
r=-1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{13}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\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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L11a170

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L11a172