L10a127

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L10a126.gif

L10a126

L10a128.gif

L10a128

Contents

L10a127.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a127 at Knotilus!


Link Presentations

[edit Notes on L10a127's Link Presentations]

Planar diagram presentation X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X18,10,19,9 X16,12,17,11 X20,18,11,17 X4,15,1,16 X10,20,5,19
Gauss code {1, -4, 3, -9}, {2, -1, 5, -3, 6, -10}, {7, -2, 4, -5, 9, -7, 8, -6, 10, -8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10a127 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) (w-1) (2 v w-v-w+2)}{\sqrt{u} v w} (db)
Jones polynomial -q^8+4 q^7-8 q^6+12 q^5-15 q^4+17 q^3-14 q^2+13 q-7+4 q^{-1} - q^{-2} (db)
Signature 2 (db)
HOMFLY-PT polynomial z^6 a^{-2} +z^6 a^{-4} +2 z^4 a^{-2} +2 z^4 a^{-4} -z^4 a^{-6} -z^4+z^2 a^{-2} +z^2 a^{-4} -z^2 a^{-6} -z^2- a^{-2} +1-2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2} (db)
Kauffman polynomial 2 z^9 a^{-3} +2 z^9 a^{-5} +5 z^8 a^{-2} +11 z^8 a^{-4} +6 z^8 a^{-6} +6 z^7 a^{-1} +10 z^7 a^{-3} +11 z^7 a^{-5} +7 z^7 a^{-7} -13 z^6 a^{-4} -5 z^6 a^{-6} +4 z^6 a^{-8} +4 z^6+a z^5-8 z^5 a^{-1} -20 z^5 a^{-3} -24 z^5 a^{-5} -12 z^5 a^{-7} +z^5 a^{-9} -11 z^4 a^{-2} -z^4 a^{-4} -3 z^4 a^{-6} -6 z^4 a^{-8} -7 z^4-a z^3+2 z^3 a^{-1} +7 z^3 a^{-3} +11 z^3 a^{-5} +6 z^3 a^{-7} -z^3 a^{-9} +6 z^2 a^{-2} +2 z^2 a^{-4} +2 z^2 a^{-6} +2 z^2 a^{-8} +4 z^2-z a^{-1} -z a^{-3} + a^{-2} + a^{-4} +1+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} z^{-2} - 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 \
-3-2-101234567χ
17          1-1
15         3 3
13        51 -4
11       73  4
9      85   -3
7     97    2
5    710     3
3   67      -1
1  39       6
-1 14        -3
-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}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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

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