Structure and Operations

2008-12-16T17:04:13Z

CooloCochi: deltabas

acrelvar
{{Manual TOC Sidebar}}

{{Startup Note}}



{{HelpLine|
n = 1 |
in = <nowiki>Crossings</nowiki> |
out= <nowiki>Crossings[L] returns the number of crossings of a knot/link L (in its given presentation).</nowiki>}}




{{HelpLine|
n = 2 |
in = <nowiki>PositiveCrossings</nowiki> |
out= <nowiki>PositiveCrossings[L] returns the number of positive (right handed) crossings in a knot/link L (in its given presentation).</nowiki>}}




{{HelpLine|
n = 3 |
in = <nowiki>NegativeCrossings</nowiki> |
out= <nowiki>NegativeCrossings[L] returns the number of negaitve (left handed) crossings in a knot/link L (in its given presentation).</nowiki>}}


Thus here's one tautology and one easy example:



{{InOut|
n = 4 |
in = <nowiki>Crossings /@ {Knot[0, 1], TorusKnot[11,10]}</nowiki> |
out= <nowiki>{0, 99}</nowiki>}}


And another easy example:



{{InOut|
n = 5 |
in = <nowiki>K=Knot[6, 2]; {PositiveCrossings[K], NegativeCrossings[K]}</nowiki> |
out= <nowiki>{2, 4}</nowiki>}}




{{HelpLine|
n = 6 |
in = <nowiki>PositiveQ</nowiki> |
out= <nowiki>PositiveQ[xing] returns True if xing is a positive (right handed) crossing and False if it is negative (left handed).</nowiki>}}




{{HelpLine|
n = 7 |
in = <nowiki>NegativeQ</nowiki> |
out= <nowiki>NegativeQ[xing] returns True if xing is a negative (left handed) crossing and False if it is positive (right handed).</nowiki>}}


For example,



{{InOut|
n = 8 |
in = <nowiki>PositiveQ /@ {X[1,3,2,4], X[1,4,2,3], Xp[1,3,2,4], Xp[1,4,2,3]}</nowiki> |
out= <nowiki>{False, True, True, True}</nowiki>}}




{{HelpLine|
n = 9 |
in = <nowiki>ConnectedSum</nowiki> |
out= <nowiki>ConnectedSum[K1, K2] represents the connected sum of the knots K1 and K2 (ConnectedSum may not work with links).</nowiki>}}


The connected sum <math>K=4_1\#4_1</math> of the knot [[4_1]] with itself has 8 crossings (unsurprisingly):



{{InOut|
n = 10 |
in = <nowiki>K = ConnectedSum[Knot[4,1], Knot[4,1]]</nowiki> |
out= <nowiki>ConnectedSum[Knot[4, 1], Knot[4, 1]]</nowiki>}}




{{InOut|
n = 11 |
in = <nowiki>Crossings[K]</nowiki> |
out= <nowiki>8</nowiki>}}


It is also nice to know that, as expected, the Jones polynomial of <math>K</math> is the square of the Jones polynomial of [[4_1]]:



{{InOut|
n = 12 |
in = <nowiki>Jones[K][q] == Expand[Jones[Knot[4,1]][q]^2]</nowiki> |
out= <nowiki>True</nowiki>}}


{{Knot Image Pair|4_1|gif|8_9|gif}}

It is less nice to know that the Jones polynomial cannot tell <math>K</math> apart from the knot [[8_9]]:



{{InOut|
n = 13 |
in = <nowiki>Jones[K][q] == Jones[Knot[8,9]][q]</nowiki> |
out= <nowiki>True</nowiki>}}


But <math>K=4_1\#4_1</math> isn't equivalent to [[8_9]]; indeed, their Alexander polynomials are different:



{{InOut|
n = 14 |
in = <nowiki>{Alexander[K][t], Alexander[Knot[8,9]][t]}</nowiki> |
out= <nowiki> -2 6 2 -3 3 5 2 3
{11 + t - - - 6 t + t , 7 - t + -- - - - 5 t + 3 t - t }
t 2 t
t</nowiki>}}

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