Structure and Operations

(For In[1] see Setup)

In[2]:= ?Crossings Crossings[L] returns the number of crossings of a knot/link L (in its given presentation).

In[3]:= ?PositiveCrossings PositiveCrossings[L] returns the number of positive (right handed) crossings in a knot/link L (in its given presentation).

In[4]:= ?NegativeCrossings NegativeCrossings[L] returns the number of negaitve (left handed) crossings in a knot/link L (in its given presentation).

Thus here's one tautology and one easy example:

In[5]:=	Crossings /@ {Knot[0, 1], TorusKnot[11,10]}
Out[5]=	{0, 99}

And another easy example:

In[6]:=	K=Knot[6, 2]; {PositiveCrossings[K], NegativeCrossings[K]}
Out[6]=	{2, 4}

In[7]:= ?PositiveQ PositiveQ[xing] returns True if xing is a positive (right handed) crossing and False if it is negative (left handed).

In[8]:= ?NegativeQ NegativeQ[xing] returns True if xing is a negative (left handed) crossing and False if it is positive (right handed).

For example,

In[9]:=	PositiveQ /@ {X[1,3,2,4], X[1,4,2,3], Xp[1,3,2,4], Xp[1,4,2,3]}
Out[9]=	{False, True, True, True}

In[10]:= ?ConnectedSum ConnectedSum[K1, K2] represents the connected sum of the knots K1 and K2 (ConnectedSum may not work with links).

The connected sum $K=4_{1}\#4_{1}$ of the knot 4_1 with itself has 8 crossings (unsurprisingly):

In[11]:=	K = ConnectedSum[Knot[4,1], Knot[4,1]]
Out[11]=	ConnectedSum[Knot[4, 1], Knot[4, 1]]

In[12]:=	Crossings[K]
Out[12]=	8

It is also nice to know that, as expected, the Jones polynomial of $K$ is the square of the Jones polynomial of 4_1:

In[13]:=	Jones[K][q] == Expand[Jones[Knot[4,1]][q]^2]
Out[13]=	True

It is less nice to know that the Jones polynomial cannot tell $K$ apart from the knot 8_9:

In[14]:=	Jones[K][q] == Jones[Knot[8,9]][q]
Out[14]=	True

But $K=4_{1}\#4_{1}$ isn't equivalent to 8_9; indeed, their Alexander polynomials are different:

In[15]:=	{Alexander[K][t], Alexander[Knot[8,9]][t]}
Out[15]=	-2 6 2 -3 3 5 2 3 {11 + t - - - 6 t + t , 7 - t + -- - - - 5 t + 3 t - t } t 2 t t