Template:Rolfsen Knot Page

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Visit [http://newweb.cecm.sfu.ca/cgi-bin/KnotPlot/KnotServer/kserver?ncomp=1&ncross={Template:N Rolfsen Knot Page Quick Notes&id={{{k}}} Rolfsen Knot Page's page] at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit [{{{KnotilusURL}}} Rolfsen Knot Page's page] at Knotilus! Visit [http://www.math.toronto.edu/~drorbn/KAtlas/Knots/{Template:N Rolfsen Knot Page Quick Notes.{{{k}}}.html Rolfsen Knot Page's page] at the original Knot Atlas!|k={{{k}}}|KnotilusURL={{{KnotilusURL}}}}} Rolfsen Knot Page Further Notes and Views Knot presentations Planar diagram presentation Data:Rolfsen Knot Page/PD Presentation Gauss code Data:Rolfsen Knot Page/Gauss Code Dowker-Thistlethwaite code Data:Rolfsen Knot Page/DT Code Conway Notation Data:Rolfsen Knot Page/Conway Notation Minimum Braid Representative: {{{braid_table}}} Length is {{{braid_crossings}}}, width is {{{braid_width}}}. Braid index is {{{braid_index}}}. A Morse Link Presentation: Three dimensional invariants Symmetry type Data:Rolfsen Knot Page/Symmetry Type Unknotting number Data:Rolfsen Knot Page/Unknotting Number 3-genus Data:Rolfsen Knot Page/3-Genus Bridge index Missing Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:Rolfsen Knot Page/ThurstonBennequinNumber Hyperbolic Volume Data:Rolfsen Knot Page/HyperbolicVolume A-Polynomial See Data:Rolfsen Knot Page/A-polynomial [edit Notes for Rolfsen Knot Page's three dimensional invariants] Four dimensional invariants Smooth 4 genus Missing Topological 4 genus Missing Concordance genus Missing Rasmussen s-Invariant Missing [edit Notes for Rolfsen Knot Page's four dimensional invariants] Polynomial invariants Alexander polynomial Data:Rolfsen Knot Page/Alexander Polynomial Conway polynomial Data:Rolfsen Knot Page/Conway Polynomial 2nd Alexander ideal (db, data sources) Data:Rolfsen Knot Page/2nd AlexanderIdeal Determinant and Signature { Data:Rolfsen Knot Page/Determinant, Data:Rolfsen Knot Page/Signature } Jones polynomial Data:Rolfsen Knot Page/Jones Polynomial HOMFLY-PT polynomial (db, data sources) Data:Rolfsen Knot Page/HOMFLYPT Polynomial Kauffman polynomial (db, data sources) Data:Rolfsen Knot Page/Kauffman Polynomial The A2 invariant Data:Rolfsen Knot Page/QuantumInvariant/A2/1,0 The G2 invariant Data:Rolfsen Knot Page/QuantumInvariant/G2/1,0 Further Quantum Invariants Rolfsen Knot Page Quantum Invariants. Computer Talk The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb. (The path below may be different on your system, and possibly also the KnotTheory` date) In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory` Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory. In[3]:= K = Knot["Rolfsen Knot Page"]; In[4]:= Alexander[K][t] KnotTheory::loading: Loading precomputed data in PD4Knots`. Out[4]= Data:Rolfsen Knot Page/Alexander Polynomial In[5]:= Conway[K][z] Out[5]= Data:Rolfsen Knot Page/Conway Polynomial In[6]:= Alexander[K, 2][t] KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005. Out[6]= Data:Rolfsen Knot Page/2nd AlexanderIdeal In[7]:= {KnotDet[K], KnotSignature[K]} Out[7]= { Data:Rolfsen Knot Page/Determinant, Data:Rolfsen Knot Page/Signature } In[8]:= Jones[K][q] KnotTheory::loading: Loading precomputed data in Jones4Knots`. Out[8]= Data:Rolfsen Knot Page/Jones Polynomial In[9]:= HOMFLYPT[K][a, z] KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison. Out[9]= Data:Rolfsen Knot Page/HOMFLYPT Polynomial In[10]:= Kauffman[K][a, z] KnotTheory::loading: Loading precomputed data in Kauffman4Knots`. Out[10]= Data:Rolfsen Knot Page/Kauffman Polynomial "Similar" Knots (within the Atlas) Same Alexander/Conway Polynomial: {{{{same_alexander}}}} Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {{{{same_jones}}}} Computer Talk The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. (The path below may be different on your system, and possibly also the KnotTheory` date) In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory` Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory. In[3]:= K = Knot["Rolfsen Knot Page"]; In[4]:= {A = Alexander[K][t], J = Jones[K][q]} KnotTheory::loading: Loading precomputed data in PD4Knots`. KnotTheory::loading: Loading precomputed data in Jones4Knots`. Out[4]= { Data:Rolfsen Knot Page/Alexander Polynomial, Data:Rolfsen Knot Page/Jones Polynomial } In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K] KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`. KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005. Out[5]= {{{{same_alexander}}}} In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ] KnotTheory::loading: Loading precomputed data in Jones4Knots11`. Out[6]= {{{{same_jones}}}} Vassiliev invariants V2 and V3: (Data:Rolfsen Knot Page/V 2, Data:Rolfsen Knot Page/V 3) V2,1 through V6,9: V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 Data:Rolfsen Knot Page/V 2,1 Data:Rolfsen Knot Page/V 3,1 Data:Rolfsen Knot Page/V 4,1 Data:Rolfsen Knot Page/V 4,2 Data:Rolfsen Knot Page/V 4,3 Data:Rolfsen Knot Page/V 5,1 Data:Rolfsen Knot Page/V 5,2 Data:Rolfsen Knot Page/V 5,3 Data:Rolfsen Knot Page/V 5,4 Data:Rolfsen Knot Page/V 6,1 Data:Rolfsen Knot Page/V 6,2 Data:Rolfsen Knot Page/V 6,3 Data:Rolfsen Knot Page/V 6,4 Data:Rolfsen Knot Page/V 6,5 Data:Rolfsen Knot Page/V 6,6 Data:Rolfsen Knot Page/V 6,7 Data:Rolfsen Knot Page/V 6,8 Data:Rolfsen Knot Page/V 6,9 V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3. Khovanov Homology The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$Data:Rolfsen Knot Page/Signature is the signature of Rolfsen Knot Page. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:Rolfsen Knot Page/KhovanovTable Integral Khovanov Homology (db, data source)    Data:Rolfsen Knot Page/Integral Khovanov Homology The Coloured Jones Polynomials The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)  $n$  $J_{n}$ 2 {{{coloured_jones_2}}} 3 {{{coloured_jones_3}}} 4 {{{coloured_jones_4}}} 5 {{{coloured_jones_5}}} 6 {{{coloured_jones_6}}} 7 {{{coloured_jones_7}}} Computer Talk Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session. {{{computer_talk}}} See/edit the Rolfsen_Splice_Template. Back to the top. 8_16 8_18