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7.2 Three Dimensional Invariants

In[1]

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 ${\mathbb{R}}^3$.

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 ${\mathbb{R}}^3$, 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, 9, {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 {13 - t + -- - -- - 10 t + 5 t - t , 2 t t -3 5 12 2 3 > -15 + t - -- + -- + 12 t - 5 t + t , 2 t t -3 5 12 2 3 > 17 - t + -- - -- - 12 t + 5 t - t , 2 t t -3 6 16 2 3 > 23 - t + -- - -- - 16 t + 6 t - t } 2 t t