L10n112

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L10n111.gif

L10n111

L10n113.gif

L10n113

Contents

L10n112.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10n112's Link Presentations]

Planar diagram presentation X6172 X2536 X11,19,12,18 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X15,17,16,20 X19,13,20,16 X17,9,18,12
Gauss code {1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 10}, {-7, 6, -8, 9}, {-10, 3, -9, 8}
A Braid Representative
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A Morse Link Presentation L10n112 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(1) t(2)+t(1) t(3) t(2)-t(3) t(2)+t(1) t(4) t(2)-t(1) t(3) t(4) t(2)+t(3) t(4) t(2)+2 t(1) t(5) t(2)-t(1) t(3) t(5) t(2)+t(3) t(5) t(2)-t(1) t(4) t(5) t(2)-t(5) t(2)+t(3)-t(1) t(4)+t(1) t(3) t(4)-2 t(3) t(4)+t(4)-t(1) t(5)-t(3) t(5)+t(1) t(4) t(5)+t(3) t(4) t(5)-t(4) t(5)+t(5)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} \sqrt{t(4)} \sqrt{t(5)}} (db)
Jones polynomial q^9-q^8+6 q^7-5 q^6+11 q^5-5 q^4+10 q^3-5 q^2+4 q (db)
Signature 2 (db)
HOMFLY-PT polynomial  a^{-10} z^{-4} + a^{-10} z^{-2} -4 a^{-8} z^{-4} -7 a^{-8} z^{-2} -4 a^{-8} +6 a^{-6} z^{-4} +6 z^2 a^{-6} +15 a^{-6} z^{-2} +14 a^{-6} -3 z^4 a^{-4} -4 a^{-4} z^{-4} -10 z^2 a^{-4} -13 a^{-4} z^{-2} -16 a^{-4} + a^{-2} z^{-4} +4 z^2 a^{-2} +4 a^{-2} z^{-2} +6 a^{-2} (db)
Kauffman polynomial z^8 a^{-6} +z^8 a^{-8} +5 z^7 a^{-5} +6 z^7 a^{-7} +z^7 a^{-9} +10 z^6 a^{-4} +12 z^6 a^{-6} +3 z^6 a^{-8} +z^6 a^{-10} +6 z^5 a^{-3} -6 z^5 a^{-7} -25 z^4 a^{-4} -39 z^4 a^{-6} -19 z^4 a^{-8} -5 z^4 a^{-10} -10 z^3 a^{-3} -30 z^3 a^{-5} -30 z^3 a^{-7} -10 z^3 a^{-9} +10 z^2 a^{-2} +30 z^2 a^{-4} +40 z^2 a^{-6} +30 z^2 a^{-8} +10 z^2 a^{-10} +20 z a^{-3} +55 z a^{-5} +55 z a^{-7} +20 z a^{-9} -10 a^{-2} -25 a^{-4} -31 a^{-6} -25 a^{-8} -10 a^{-10} -15 a^{-3} z^{-1} -41 a^{-5} z^{-1} -41 a^{-7} z^{-1} -15 a^{-9} z^{-1} +5 a^{-2} z^{-2} +14 a^{-4} z^{-2} +18 a^{-6} z^{-2} +14 a^{-8} z^{-2} +5 a^{-10} z^{-2} +4 a^{-3} z^{-3} +12 a^{-5} z^{-3} +12 a^{-7} z^{-3} +4 a^{-9} z^{-3} - a^{-2} z^{-4} -4 a^{-4} z^{-4} -6 a^{-6} z^{-4} -4 a^{-8} z^{-4} - a^{-10} 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 \
012345678χ
19        11
17       110
15      5  5
13      1  1
11    115   6
9   410    6
7  61     5
51 4      5
356       -1
14        4
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3 i=5
r=0 {\mathbb Z}^{4} {\mathbb Z}^{5} {\mathbb Z}
r=1 {\mathbb Z}^{6}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2 {\mathbb Z}^{11}
r=5 {\mathbb Z}^{5}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=7 {\mathbb Z}_2 {\mathbb Z}
r=8 {\mathbb Z} {\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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L10n111.gif

L10n111

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L10n113