Structure and Operations

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(For In[1] see Setup)

In[1]:= ?Crossings
Crossings[L] returns the number of crossings of a knot/link L (in its given presentation).
In[2]:= ?PositiveCrossings
PositiveCrossings[L] returns the number of positive (right handed) crossings in a knot/link L (in its given presentation).
In[3]:= ?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[4]:= Crossings /@ {Knot[0, 1], TorusKnot[11,10]}
Out[4]= {0, 99}

And another easy example:

In[5]:= K=Knot[6, 2]; {PositiveCrossings[K], NegativeCrossings[K]}
Out[5]= {2, 4}
In[6]:= ?PositiveQ
PositiveQ[xing] returns True if xing is a positive (right handed) crossing and False if it is negative (left handed).
In[7]:= ?NegativeQ
NegativeQ[xing] returns True if xing is a negative (left handed) crossing and False if it is positive (right handed).

For example,

In[8]:= PositiveQ /@ {X[1,3,2,4], X[1,4,2,3], Xp[1,3,2,4], Xp[1,4,2,3]}
Out[8]= {False, True, True, True}
In[9]:= ?ConnectedSum
ConnectedSum[K1, K2] represents the connected sum of the knots K1 and K2 (ConnectedSum may not work with links).

The connected sum of the knot 4_1 with itself has 8 crossings (unsurprisingly):

In[10]:= K = ConnectedSum[Knot[4,1], Knot[4,1]]
Out[10]= ConnectedSum[Knot[4, 1], Knot[4, 1]]
In[11]:= Crossings[K]
Out[11]= 8

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


In[12]:= Jones[K][q] == Expand[Jones[Knot[4,1]][q]^2]
Out[12]= True

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

In[13]:= Jones[K][q] == Jones[Knot[8,9]][q]
Out[13]= True

But isn't equivalent to 8_9; indeed, their Alexander polynomials are different:

In[14]:= {Alexander[K][t], Alexander[Knot[8,9]][t]}
Out[14]= -2 6 2 -3 3 5 2 3 {11 + t - - - 6 t + t , 7 - t + -- - - - 5 t + 3 t - t } t 2 t t