L11a508

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L11a507.gif

L11a507

L11a509.gif

L11a509

Contents

L11a508.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a508's Link Presentations]

Planar diagram presentation X8192 X18,12,19,11 X10,4,11,3 X2,20,3,19 X16,8,17,7 X20,9,21,10 X12,18,7,17 X4,21,5,22 X14,6,15,5 X22,15,13,16 X6,14,1,13
Gauss code {1, -4, 3, -8, 9, -11}, {5, -1, 6, -3, 2, -7}, {11, -9, 10, -5, 7, -2, 4, -6, 8, -10}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
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A Morse Link Presentation L11a508 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(w-1) \left(u^2 v^2 w+u^2 \left(-v^2\right)+u^2 v w^2-2 u^2 v w+u^2 v-u^2 w^2+u^2 w-2 u v^2 w+2 u v^2-2 u v w^2+3 u v w-2 u v+2 u w^2-2 u w+v^2 w-v^2+v w^2-2 v w+v-w^2+w\right)}{u v w^{3/2}} (db)
Jones polynomial -q^7+4 q^6-7 q^5+13 q^4+ q^{-4} -16 q^3-4 q^{-3} +20 q^2+8 q^{-2} -19 q-13 q^{-1} +18 (db)
Signature 0 (db)
HOMFLY-PT polynomial -z^6 a^{-2} -z^6+a^2 z^4-z^4 a^{-2} +2 z^4 a^{-4} -2 z^4+a^2 z^2+z^2 a^{-2} +2 z^2 a^{-4} -z^2 a^{-6} -2 z^2+2 a^{-2} -2 a^{-4} + a^{-2} z^{-2} -2 a^{-4} z^{-2} + a^{-6} z^{-2} (db)
Kauffman polynomial z^7 a^{-7} -3 z^5 a^{-7} +2 z^3 a^{-7} +4 z^8 a^{-6} -15 z^6 a^{-6} +17 z^4 a^{-6} -5 z^2 a^{-6} + a^{-6} z^{-2} -2 a^{-6} +5 z^9 a^{-5} -15 z^7 a^{-5} +10 z^5 a^{-5} +2 z a^{-5} -2 a^{-5} z^{-1} +2 z^{10} a^{-4} +7 z^8 a^{-4} -41 z^6 a^{-4} +a^4 z^4+47 z^4 a^{-4} -14 z^2 a^{-4} +2 a^{-4} z^{-2} -3 a^{-4} +12 z^9 a^{-3} -30 z^7 a^{-3} +4 a^3 z^5+14 z^5 a^{-3} -2 a^3 z^3+2 z^3 a^{-3} +2 z a^{-3} -2 a^{-3} z^{-1} +2 z^{10} a^{-2} +14 z^8 a^{-2} +8 a^2 z^6-49 z^6 a^{-2} -7 a^2 z^4+42 z^4 a^{-2} +2 a^2 z^2-11 z^2 a^{-2} + a^{-2} z^{-2} -2 a^{-2} +7 z^9 a^{-1} +11 a z^7-3 z^7 a^{-1} -13 a z^5-16 z^5 a^{-1} +4 a z^3+10 z^3 a^{-1} +11 z^8-15 z^6+4 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 \
-4-3-2-101234567χ
15           1-1
13          3 3
11         41 -3
9        93  6
7       85   -3
5      128    4
3     910     1
1    910      -1
-1   510       5
-3  38        -5
-5 15         4
-7 3          -3
-91           1
Integral Khovanov Homology

(db, data source)

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

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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L11a507.gif

L11a507

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L11a509