L11a486

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L11a485.gif

L11a485

L11a487.gif

L11a487

Contents

L11a486.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a486 at Knotilus!


Link Presentations

[edit Notes on L11a486's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 X12,6,13,5 X8493 X22,14,19,13 X20,10,21,9 X10,20,11,19 X14,22,15,21 X18,12,5,11 X2,16,3,15
Gauss code {1, -11, 5, -3}, {8, -7, 9, -6}, {4, -1, 2, -5, 7, -8, 10, -4, 6, -9, 11, -2, 3, -10}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
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A Morse Link Presentation L11a486 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(3)-1)^2 \left(t(3)^3+2 t(2) t(3)^2-2 t(3)^2-2 t(2) t(3)+2 t(3)+t(2)\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{5/2}} (db)
Jones polynomial q^9-4 q^8+10 q^7-17 q^6+23 q^5-25 q^4+27 q^3-22 q^2+17 q-9+4 q^{-1} - q^{-2} (db)
Signature 2 (db)
HOMFLY-PT polynomial z^2 a^{-8} + a^{-8} -3 z^4 a^{-6} -5 z^2 a^{-6} + a^{-6} z^{-2} -2 a^{-6} +2 z^6 a^{-4} +5 z^4 a^{-4} +4 z^2 a^{-4} -2 a^{-4} z^{-2} - a^{-4} +z^6 a^{-2} +z^4 a^{-2} +z^2 a^{-2} + a^{-2} z^{-2} +2 a^{-2} -z^4-z^2 (db)
Kauffman polynomial 2 z^{10} a^{-4} +2 z^{10} a^{-6} +8 z^9 a^{-3} +15 z^9 a^{-5} +7 z^9 a^{-7} +11 z^8 a^{-2} +25 z^8 a^{-4} +22 z^8 a^{-6} +8 z^8 a^{-8} +8 z^7 a^{-1} +z^7 a^{-3} -14 z^7 a^{-5} -3 z^7 a^{-7} +4 z^7 a^{-9} -16 z^6 a^{-2} -67 z^6 a^{-4} -65 z^6 a^{-6} -17 z^6 a^{-8} +z^6 a^{-10} +4 z^6+a z^5-10 z^5 a^{-1} -22 z^5 a^{-3} -23 z^5 a^{-5} -20 z^5 a^{-7} -8 z^5 a^{-9} +12 z^4 a^{-2} +63 z^4 a^{-4} +62 z^4 a^{-6} +14 z^4 a^{-8} -2 z^4 a^{-10} -5 z^4-a z^3+5 z^3 a^{-1} +19 z^3 a^{-3} +27 z^3 a^{-5} +20 z^3 a^{-7} +6 z^3 a^{-9} -4 z^2 a^{-2} -27 z^2 a^{-4} -30 z^2 a^{-6} -8 z^2 a^{-8} +z^2 a^{-10} +2 z^2-z a^{-1} -z a^{-3} -5 z a^{-5} -7 z a^{-7} -2 z a^{-9} - a^{-2} +2 a^{-4} +5 a^{-6} +3 a^{-8} -2 a^{-3} z^{-1} -2 a^{-5} z^{-1} + a^{-2} z^{-2} +2 a^{-4} z^{-2} + a^{-6} z^{-2} (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 \
-3-2-1012345678χ
19           11
17          3 -3
15         71 6
13        103  -7
11       137   6
9      1412    -2
7     1311     2
5    914      5
3   813       -5
1  311        8
-1 16         -5
-3 3          3
-51           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=4 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{13}
r=5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=8 {\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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L11a485.gif

L11a485

L11a487.gif

L11a487