T(5,2)

T(3,2)

T(7,2)

Visit [[[:Template:KnotilusURL]] T(5,2)'s page] at Knotilus! Visit T(5,2)'s page at the original Knot Atlas!

An interlaced pentagram, this is known variously as the "Cinquefoil Knot", after certain herbs and shrubs of the rose family which have 5-lobed leaves and 5-petaled flowers (see e.g. [4]), as the "Pentafoil Knot" (visit Bert Jagers' pentafoil page), as the "Double Overhand Knot", as 5_1, or finally as the torus knot T(5,2). When taken off the post the strangle knot (hitch) of practical knot tying deforms to 5_1

A kolam of a 2x3 dot array	The VISA Interlink Logo [1]	Version of the US bicentennial emblem
A pentagonal table by Bob Mackay [2]	The Utah State Parks logo	As impossible object ("Penrose" pentagram)
Folded ribbon which is single-sided (more complex version of Möbius Strip).	Non-pentagonal shape.	Pentagram of circles.
Alternate pentagram of intersecting circles.	3D-looking rendition.	Partial view of US bicentennial logo on a shirt seen in Lisboa [3]
Non-prime knot with two 5_1 configurations on a closed loop.	Knotted epitrochoid	Sum of two 5_1s, Vienna, orthodox church

This sentence was last edited by Dror. Sometime later, Scott added this sentence.

Knot presentations

Planar diagram presentation	X3948 X9,5,10,4 X5,1,6,10 X1726 X7382
Gauss code	-4, 5, -1, 2, -3, 4, -5, 1, -2, 3
Dowker-Thistlethwaite code	6 8 10 2 4
Conway Notation	Data:T(5,2)/Conway Notation

Polynomial invariants

Alexander polynomial	$t^{2}-t+1-t^{-1}+t^{-2}$
Conway polynomial	$z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 5, 4 }
Jones polynomial	$-q^{7}+q^{6}-q^{5}+q^{4}+q^{2}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{-4}+4z^{2}a^{-4}-z^{2}a^{-6}+3a^{-4}-2a^{-6}$
Kauffman polynomial (db, data sources)	$z^{4}a^{-4}+z^{4}a^{-6}+z^{3}a^{-5}+z^{3}a^{-7}-4z^{2}a^{-4}-3z^{2}a^{-6}+z^{2}a^{-8}-2za^{-5}-za^{-7}+za^{-9}+3a^{-4}+2a^{-6}$
The A2 invariant	Data:T(5,2)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(5,2)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(5,2) Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["T(5,2)"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$t^{2}-t+1-t^{-1}+t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{4}+3z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 5, 4 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$-q^{7}+q^{6}-q^{5}+q^{4}+q^{2}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$z^{4}a^{-4}+4z^{2}a^{-4}-z^{2}a^{-6}+3a^{-4}-2a^{-6}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$z^{4}a^{-4}+z^{4}a^{-6}+z^{3}a^{-5}+z^{3}a^{-7}-4z^{2}a^{-4}-3z^{2}a^{-6}+z^{2}a^{-8}-2za^{-5}-za^{-7}+za^{-9}+3a^{-4}+2a^{-6}$

Vassiliev invariants

V2 and V3:

(3, 5)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 Data:T(5,2)/V 2,1 Data:T(5,2)/V 3,1 Data:T(5,2)/V 4,1 Data:T(5,2)/V 4,2 Data:T(5,2)/V 4,3 Data:T(5,2)/V 5,1 Data:T(5,2)/V 5,2 Data:T(5,2)/V 5,3 Data:T(5,2)/V 5,4 Data:T(5,2)/V 6,1 Data:T(5,2)/V 6,2 Data:T(5,2)/V 6,3 Data:T(5,2)/V 6,4 Data:T(5,2)/V 6,5 Data:T(5,2)/V 6,6 Data:T(5,2)/V 6,7 Data:T(5,2)/V 6,8 Data:T(5,2)/V 6,9

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$4 is the signature of T(5,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 χ 15           1 -1 13             0 11       1 1   0 9             0 7     1       1 5 1           1 3 1           1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=1$ $r=2$ ${\mathbb {Z} }$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=4$ ${\mathbb {Z} }$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[TorusKnot[5, 2]]
Out[2]=	5
In[3]:=	PD[TorusKnot[5, 2]]
Out[3]=	PD[X[3, 9, 4, 8], X[9, 5, 10, 4], X[5, 1, 6, 10], X[1, 7, 2, 6], X[7, 3, 8, 2]]
In[4]:=	GaussCode[TorusKnot[5, 2]]
Out[4]=	GaussCode[-4, 5, -1, 2, -3, 4, -5, 1, -2, 3]
In[5]:=	BR[TorusKnot[5, 2]]
Out[5]=	BR[2, {1, 1, 1, 1, 1}]
In[6]:=	alex = Alexander[TorusKnot[5, 2]][t]
Out[6]=	-2 1 2 1 + Alternating - ----------- - Alternating + Alternating Alternating
In[7]:=	Conway[TorusKnot[5, 2]][z]
Out[7]=	2 4 1 + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[5, 1], Knot[10, 132]}
In[9]:=	{KnotDet[TorusKnot[5, 2]], KnotSignature[TorusKnot[5, 2]]}
Out[9]=	{5, 4}
In[10]:=	J=Jones[TorusKnot[5, 2]][q]
Out[10]=	2 4 5 6 7 q + q - q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[5, 1], Knot[10, 132]}
In[12]:=	A2Invariant[TorusKnot[5, 2]][q]
Out[12]=	6 8 10 12 14 18 20 22 q + q + 2 q + q + q - q - q - q
In[13]:=	Kauffman[TorusKnot[5, 2]][a, z]
Out[13]=	2 2 2 3 3 4 4 2 3 z z 2 z z 3 z 4 z z z z z -- + -- + -- - -- - --- + -- - ---- - ---- + -- + -- + -- + -- 6 4 9 7 5 8 6 4 7 5 6 4 a a a a a a a a a a a a
In[14]:=	{Vassiliev[2][TorusKnot[5, 2]], Vassiliev[3][TorusKnot[5, 2]]}
Out[14]=	{0, 5}
In[15]:=	Kh[TorusKnot[5, 2]][q, t]
Out[15]=	3 5 2 7 3 11 4 11 q + q + Alternating q + Alternating q + Alternating q + 5 15 Alternating q