Three Dimensional Invariants

KnotTheory`/Navigation  The Mathematica Package KnotTheory` Acknowledgement Setup Naming and Enumeration Presentations Planar Diagrams Some further details Gauss Codes DT (Dowker-Thistlethwaite) Codes Braid Representatives MorseLink Presentations Arc Presentations Conway Notation Graphical Input Graphical Output Drawing Planar Diagrams How does it work? Drawing MorseLink Presentations Drawing Braids Structure and Operations Invariants Invariants from Braid Theory Three Dimensional Invariants The Alexander-Conway Polynomial The Multivariable Alexander Polynomial The Determinant and the Signature The Jones Polynomial How is the Jones polynomial computed? The Coloured Jones Polynomials The A2 Invariant Quantum knot invariants The HOMFLY-PT Polynomial The Kauffman Polynomial Finite Type (Vassiliev) Invariants Khovanov Homology Heegaard Floer Knot Homology R-Matrix Invariants Extras Included with KnotTheory` Drawing with TubePlot Standalone_TubePlot Using the LinKnot package WikiLink QuantumGroups` Lightly Documented Features A Sample KnotTheory` Session Further Usage Example Prime Links with a Non-Prime Component "Rubberband" Brunnian Links Identifying Knots within a List Cabling Burau's Theorem Further Knot Theory Software KnotPlot Knotscape Xknots Extending/Modifying KnotTheory` Bugs How to Edit this Manual...

(For In[1] see Setup)

In[2]:= ?SymmetryType SymmetryType[K] returns the symmetry type of the knot K, if known to KnotTheory`. The possible types are: Reversible, FullyAmphicheiral, NegativeAmphicheiral and Chiral.

In[3]:= SymmetryType::about The symmetry type data known to KnotTheory` is taken from Charles Livingston's "Table of Knot Invariants", http://www.indiana.edu/~knotinfo/.

The unknotting number of a knot $K$ is the minimal number of crossing changes needed in order to unknot $K$.

In[4]:= ?UnknottingNumber UnknottingNumber[K] returns the unknotting number of the knot K, if known to KnotTheory`. If only a range of possible values is known, a list of the form {min, max} is returned.

In[5]:= UnknottingNumber::about The unknotting numbers of torus knots are due to ???. All other unknotting numbers known to KnotTheory` are taken from Charles Livingston's "Table of Knot Invariants", http://www.indiana.edu/~knotinfo/.

Of the 512 knots whose unknotting number is known to KnotTheory`, 197 have unknotting number 1, 247 have unknotting number 2, 54 have unknotting number 3, 12 have unknotting number 4 and 1 has unknotting number 5:

In[6]:=	Plus @@ u /@ Cases[UnknottingNumber /@ AllKnots[], _Integer]
Out[6]=	u[0] + 197 u[1] + 247 u[2] + 54 u[3] + 12 u[4] + u[5]

There are 4 knots with up to 9 crossings whose unknotting number is unknown:

In[7]:=	Select[AllKnots[], Crossings[#] <= 9 && Head[UnknottingNumber[#]] === List &]
Out[7]=	{Knot[9, 10], Knot[9, 13], Knot[9, 35], Knot[9, 38]}

In[8]:= ?ThreeGenus ThreeGenus[K] returns the 3-genus of the knot K, if known to KnotTheory`.

In[9]:= ThreeGenus::about The 3-genus data known to KnotTheory` is taken from Charles Livingston's "Table of Knot Invariants", http://www.indiana.edu/~knotinfo/.

The bridge index' of a knot $K$ is the minimal number of local maxima (or local minima) in a generic smooth embedding of $K$ in ${\mathbf {R} }^{3}$.

In[10]:= ?BridgeIndex BridgeIndex[K] returns the bridge index of the knot K, if known to KnotTheory`.

In[11]:= BridgeIndex::about The bridge index data known to KnotTheory` is taken from Charles Livingston's "Table of Knot Invariants", http://www.indiana.edu/~knotinfo/.

An often studied class of knots is the class of 2-bridge knots, knots whose bridge index is 2. Of the 49 9-crossings knots, 24 are 2-bridge:

In[12]:=	Select[AllKnots[], Crossings[#] == 9 && BridgeIndex[#] == 2 &]
Out[12]=	{Knot[9, 1], Knot[9, 2], Knot[9, 3], Knot[9, 4], Knot[9, 5], Knot[9, 6], Knot[9, 7], Knot[9, 8], Knot[9, 9], Knot[9, 10], Knot[9, 11], Knot[9, 12], Knot[9, 13], Knot[9, 14], Knot[9, 15], Knot[9, 17], Knot[9, 18], Knot[9, 19], Knot[9, 20], Knot[9, 21], Knot[9, 23], Knot[9, 26], Knot[9, 27], Knot[9, 31]}

The super bridge index of a knot $K$ is the minimal number, in a generic smooth embedding of $K$ in ${\mathbf {R} }^{3}$, of the maximal number of local maxima (or local minima) in a rigid rotation of that projection.

In[13]:= ?SuperBridgeIndex SuperBridgeIndex[K] returns the super bridge index of the knot K, if known to KnotTheory`. If only a range of possible values is known, a list of the form {min, max} is returned.

In[14]:= SuperBridgeIndex::about The super bridge index data known to KnotTheory` is taken from Charles Livingston's "Table of Knot Invariants", http://www.indiana.edu/~knotinfo/.

In[15]:= ?NakanishiIndex NakanishiIndex[K] returns the Nakanishi index of the knot K, if known to KnotTheory`.

In[16]:= NakanishiIndex::about The Nakanishi index data known to KnotTheory` is taken from Charles Livingston's "Table of Knot Invariants", http://www.indiana.edu/~knotinfo/.

In[17]:=

Profile[K_] := Profile[SymmetryType[K], UnknottingNumber[K], ThreeGenus[K], BridgeIndex[K], SuperBridgeIndex[K], NakanishiIndex[K]]

In[18]:=	Profile[Knot[9,24]]
Out[18]=	Profile[Reversible, 1, 3, 3, {4, 6}, 1]

In[19]:=	Ks = Select[AllKnots[], (Crossings[#] == 9 && Profile[#]==Profile[Knot[9,24]])&]
Out[19]=	{Knot[9, 24], Knot[9, 28], Knot[9, 30], Knot[9, 34]}

In[20]:=	Alexander[#][t]& /@ Ks
Out[20]=	-3 5 10 2 3 -3 5 12 2 3 {13 - t + -- - -- - 10 t + 5 t - t , -15 + t - -- + -- + 12 t - 5 t + t , 2 t 2 t t t -3 5 12 2 3 -3 6 16 2 3 17 - t + -- - -- - 12 t + 5 t - t , 23 - t + -- - -- - 16 t + 6 t - t } 2 t 2 t t t