L11a500

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L11a499.gif

L11a499

L11a501.gif

L11a501

Contents

L11a500.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a500's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X22,14,19,13 X20,8,21,7 X10,20,11,19 X16,10,17,9 X14,18,15,17 X8,16,9,15 X18,22,5,21 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {5, -4, 9, -3}, {10, -1, 4, -8, 6, -5, 11, -2, 3, -7, 8, -6, 7, -9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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A Morse Link Presentation L11a500 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(w-1) \left(2 u v w^3-4 u v w^2+3 u v w-u v-2 u w^3+4 u w^2-4 u w-4 v w^3+4 v w^2-2 v w-w^4+3 w^3-4 w^2+2 w\right)}{\sqrt{u} \sqrt{v} w^{5/2}} (db)
Jones polynomial -q^8+5 q^7-12 q^6+18 q^5-23 q^4+27 q^3-25 q^2+22 q-14+9 q^{-1} -3 q^{-2} + q^{-3} (db)
Signature 2 (db)
HOMFLY-PT polynomial z^6 a^{-2} +z^6 a^{-4} -z^4 a^{-2} +z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2-7 z^2 a^{-2} +3 z^2 a^{-4} -z^2+a^2-10 a^{-2} +7 a^{-4} - a^{-6} +3-5 a^{-2} z^{-2} +4 a^{-4} z^{-2} - a^{-6} z^{-2} +2 z^{-2} (db)
Kauffman polynomial z^5 a^{-9} +5 z^6 a^{-8} -3 z^4 a^{-8} +12 z^7 a^{-7} -16 z^5 a^{-7} +5 z^3 a^{-7} -2 z a^{-7} + a^{-7} z^{-1} +15 z^8 a^{-6} -23 z^6 a^{-6} +11 z^4 a^{-6} -4 z^2 a^{-6} - a^{-6} z^{-2} +2 a^{-6} +9 z^9 a^{-5} +3 z^7 a^{-5} -30 z^5 a^{-5} +26 z^3 a^{-5} -15 z a^{-5} +5 a^{-5} z^{-1} +2 z^{10} a^{-4} +25 z^8 a^{-4} -60 z^6 a^{-4} +47 z^4 a^{-4} -25 z^2 a^{-4} -4 a^{-4} z^{-2} +14 a^{-4} +14 z^9 a^{-3} -11 z^7 a^{-3} -31 z^5 a^{-3} +52 z^3 a^{-3} -33 z a^{-3} +9 a^{-3} z^{-1} +2 z^{10} a^{-2} +16 z^8 a^{-2} +a^2 z^6-47 z^6 a^{-2} -3 a^2 z^4+53 z^4 a^{-2} +3 a^2 z^2-40 z^2 a^{-2} -5 a^{-2} z^{-2} -a^2+21 a^{-2} +5 z^9 a^{-1} +3 a z^7+z^7 a^{-1} -6 a z^5-24 z^5 a^{-1} +3 a z^3+34 z^3 a^{-1} -20 z a^{-1} +5 a^{-1} z^{-1} +6 z^8-14 z^6+17 z^4-16 z^2-2 z^{-2} +9 (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
17           1-1
15          4 4
13         81 -7
11        104  6
9       149   -5
7      139    4
5     1214     2
3    1013      -3
1   614       8
-1  38        -5
-3  6         6
-513          -2
-71           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-4 {\mathbb Z} {\mathbb Z}
r=-3 {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=2 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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L11a499.gif

L11a499

L11a501.gif

L11a501