T(5,4)

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Further "T(5,4)" views

Planar Diagram: PD[X[17, 25, 18, 24], X[10, 26, 11, 25], X[3, 27, 4, 26], X[11, 19, 12, 18],

X[4, 20, 5, 19], X[27, 21, 28, 20], X[5, 13, 6, 12], X[28, 14, 29, 13], 
X[21, 15, 22, 14], X[29, 7, 30, 6], X[22, 8, 23, 7], X[15, 9, 16, 8], 
X[23, 1, 24, 30], X[16, 2, 17, 1], X[9, 3, 10, 2]]

<a href="../Manual/TubePlot.html"><img src="m.n_240.jpg" border=0 alt="T(m,n)"> TubePlot</a>

The m*(-1 + n)-Crossing Torus Knot T(m,n) Include["$knotaka.html"] Visit <a class=external href="KnotilusURL[GaussCode[PD[TorusKnot[m, n]]]]">T(m,n)'s page</a> at <a class=external href="http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/html/start.html">Kno tilus</a>! <a href="../Manual/Acknowledgement.html">Acknowledgement</a>

<a href="../Manual/GaussCode.html">Gauss Code</a>:

{PD[TorusKnot[m, n]]}

<a href="../Manual/BR.html">Braid Representative</a>:

BraidPlot[CollapseBraid[BR[TorusKnot[m, n]]], Mode -> "HTML"]

<a href="../Manual/AlexanderConway.html">Alexander Polynomial</a>:

PolyPrint[1, t]

<a href="../Manual/AlexanderConway.html">Conway Polynomial</a>:

PolyPrint[1, z]

Other knots with the same <a href="../Manual/AlexanderConway.html">Alexander/Conway Polynomial</a>:

{StringJoin[ToString[Knot[0, 1], FormatType -> HTMLForm], ", ", ToString[Knot[11, NonAlternating, 34], FormatType -> HTMLForm], ", ", ToString[Knot[11, NonAlternating, 42], FormatType -> HTMLForm], ", "]...}

<a href="../Manual/DetAndSignature.html">Determinant and Signature</a>:

{1, 0}

<a href="../Manual/Jones.html">Jones Polynomial</a>:

PolyPrint[-((Sqrt[q]*TorusKnot[m, n])/(1 + q)), q]

Other knots (up to mirrors) with the same <a href="../Manual/Jones.html">Jones Polynomial</a>:

{""...}

Include["ColouredJones.mhtml"]

<a href="../Manual/A2Invariant.html">A2 (sl(3)) Invariant</a>:

PolyPrint[TorusKnot[m, n], q]

<a href="../Manual/Kauffman.html">Kauffman Polynomial</a>:

PolyPrint[KnotTheory`Kauffman`StateValuation[9*I, (-I)*z][1/4]/ ((-1)^(PD[TorusKnot[m, n]]/4)*3^PD[TorusKnot[m, n]]) + KnotTheory`Kauffman`StateValuation[9*I, (-I)*z][ Flatten[KnotTheory`Kauffman`Decorate /@ #1] & ]/ ((-1)^(PD[TorusKnot[m, n]]/4)*3^PD[TorusKnot[m, n]]) + KnotTheory`Kauffman`StateValuation[9*I, (-I)*z][ {KnotTheory`Kauffman`State[PD[TorusKnot[m, n]]]}]/ ((-1)^(PD[TorusKnot[m, n]]/4)*3^PD[TorusKnot[m, n]]), {9, z}]

<a href="../Manual/Vassiliev.html">V2 and V3, the type 2 and 3 Vassiliev invariants</a>:

{0, 0}

<a href="../Manual/KhovanovHomology.html">Khovanov Homology</a>. The coefficients of the monomials t^rq^j are shown, along with their alternating sums χ (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=0 is the signature of T(m,n). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

 TabularKh[$Failed[q, t], {1, -1}]

ComputerTalkHeader

GraphicsBox["`1`.`2`_240.jpg", "TubePlot[TorusKnot[`1`, `2`]]", m, n] InOut["Crossings[``]", TorusKnot[m, n]] InOut["PD[``]", TorusKnot[m, n]] InOut["GaussCode[``]", TorusKnot[m, n]] InOut["BR[``]", TorusKnot[m, n]] InOut["alex = Alexander[``][t]", TorusKnot[m, n]] InOut["Conway[``][z]", TorusKnot[m, n]] InOut["Select[AllKnots[], (alex === Alexander[#][t])&]"] InOut["{KnotDet[`1`], KnotSignature[`1`]}", TorusKnot[m, n]] InOut["J=Jones[``][q]", TorusKnot[m, n]] InOut[

"Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]"]

Include["ColouredJonesM.mhtml"] InOut["A2Invariant[``][q]", TorusKnot[m, n]] InOut["Kauffman[``][a, z]", TorusKnot[m, n]] InOut["{Vassiliev[2][`1`], Vassiliev[3][`1`]}", TorusKnot[m, n]] InOut["Kh[``][q, t]", TorusKnot[m, n]]

---

<a href="/~drorbn/">Dror Bar-Natan</a>: <a href="../index.html">The Knot Atlas</a>: <a href="index.html">Torus Knots</a>: <a href="#top">The Torus Knot T(m,n)</a>

<a href="prevm.prevn.html"><img border=0 width=120 height=120 src="prevm.prevn_120.jpg" alt="T(prevm,prevn)"> T(prevm,prevn)</a> <a href="nextm.nextn.html"><img border=0 width=120 height=120 src="nextm.nextn_120.jpg" alt="T(nextm,nextn)"> T(nextm,nextn)</a>

</body> </html>

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Visit T(5,4)'s page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit [{{{KnotilusURL}}} T(5,4)'s page] at Knotilus!

Visit T(5,4)'s page at the original Knot Atlas!

Knot presentations

Planar diagram presentation	X17,25,18,24 X10,26,11,25 X3,27,4,26 X11,19,12,18 X4,20,5,19 X27,21,28,20 X5,13,6,12 X28,14,29,13 X21,15,22,14 X29,7,30,6 X22,8,23,7 X15,9,16,8 X23,1,24,30 X16,2,17,1 X9,3,10,2
Gauss code	14, 15, -3, -5, -7, 10, 11, 12, -15, -2, -4, 7, 8, 9, -12, -14, -1, 4, 5, 6, -9, -11, -13, 1, 2, 3, -6, -8, -10, 13
Dowker-Thistlethwaite code	16 -26 -12 22 -2 -18 28 -8 -24 4 -14 -30 10 -20 -6
Conway Notation	Data:T(5,4)/Conway Notation

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	2
Bridge index (super bridge index)	2 (4)
Nakanishi index	1

Polynomial invariants

Alexander polynomial	$t^{6}-t^{5}+t^{2}-1+t^{-2}-t^{-5}+t^{-6}$
Conway polynomial	$z^{12}+11z^{10}+44z^{8}+77z^{6}+56z^{4}+15z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 5, 8 }
Jones polynomial	$-q^{13}-q^{11}+q^{10}+q^{8}+q^{6}$
HOMFLY-PT polynomial (db, data sources)	$z^{12}a^{-12}+12z^{10}a^{-12}-z^{10}a^{-14}+55z^{8}a^{-12}-11z^{8}a^{-14}+121z^{6}a^{-12}-45z^{6}a^{-14}+z^{6}a^{-16}+133z^{4}a^{-12}-84z^{4}a^{-14}+7z^{4}a^{-16}+70z^{2}a^{-12}-70z^{2}a^{-14}+15z^{2}a^{-16}+14a^{-12}-21a^{-14}+9a^{-16}-a^{-18}$
Kauffman polynomial (db, data sources)	$z^{12}a^{-12}+z^{12}a^{-14}+z^{11}a^{-13}+z^{11}a^{-15}-12z^{10}a^{-12}-12z^{10}a^{-14}-11z^{9}a^{-13}-11z^{9}a^{-15}+55z^{8}a^{-12}+56z^{8}a^{-14}+z^{8}a^{-16}+45z^{7}a^{-13}+46z^{7}a^{-15}+z^{7}a^{-17}-121z^{6}a^{-12}-129z^{6}a^{-14}-8z^{6}a^{-16}-84z^{5}a^{-13}-91z^{5}a^{-15}-7z^{5}a^{-17}+133z^{4}a^{-12}+154z^{4}a^{-14}+21z^{4}a^{-16}+70z^{3}a^{-13}+84z^{3}a^{-15}+14z^{3}a^{-17}-70z^{2}a^{-12}-91z^{2}a^{-14}-22z^{2}a^{-16}-z^{2}a^{-18}-21za^{-13}-28za^{-15}-8za^{-17}-za^{-19}+14a^{-12}+21a^{-14}+9a^{-16}+a^{-18}$
The A2 invariant	Data:T(5,4)/QuantumInvariant/A2/1,0
The G2 invariant	Data:T(5,4)/QuantumInvariant/G2/1,0

Further Quantum Invariants

T(5,4) 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(5,4)"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{6}-t^{5}+t^{2}-1+t^{-2}-t^{-5}+t^{-6}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{12}+11z^{10}+44z^{8}+77z^{6}+56z^{4}+15z^{2}+1$

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, 8 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{13}-q^{11}+q^{10}+q^{8}+q^{6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{12}a^{-12}+12z^{10}a^{-12}-z^{10}a^{-14}+55z^{8}a^{-12}-11z^{8}a^{-14}+121z^{6}a^{-12}-45z^{6}a^{-14}+z^{6}a^{-16}+133z^{4}a^{-12}-84z^{4}a^{-14}+7z^{4}a^{-16}+70z^{2}a^{-12}-70z^{2}a^{-14}+15z^{2}a^{-16}+14a^{-12}-21a^{-14}+9a^{-16}-a^{-18}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{12}a^{-12}+z^{12}a^{-14}+z^{11}a^{-13}+z^{11}a^{-15}-12z^{10}a^{-12}-12z^{10}a^{-14}-11z^{9}a^{-13}-11z^{9}a^{-15}+55z^{8}a^{-12}+56z^{8}a^{-14}+z^{8}a^{-16}+45z^{7}a^{-13}+46z^{7}a^{-15}+z^{7}a^{-17}-121z^{6}a^{-12}-129z^{6}a^{-14}-8z^{6}a^{-16}-84z^{5}a^{-13}-91z^{5}a^{-15}-7z^{5}a^{-17}+133z^{4}a^{-12}+154z^{4}a^{-14}+21z^{4}a^{-16}+70z^{3}a^{-13}+84z^{3}a^{-15}+14z^{3}a^{-17}-70z^{2}a^{-12}-91z^{2}a^{-14}-22z^{2}a^{-16}-z^{2}a^{-18}-21za^{-13}-28za^{-15}-8za^{-17}-za^{-19}+14a^{-12}+21a^{-14}+9a^{-16}+a^{-18}$

Vassiliev invariants

V2 and V3:

(15, 50)

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(5,4)/V 2,1 Data:T(5,4)/V 3,1 Data:T(5,4)/V 4,1 Data:T(5,4)/V 4,2 Data:T(5,4)/V 4,3 Data:T(5,4)/V 5,1 Data:T(5,4)/V 5,2 Data:T(5,4)/V 5,3 Data:T(5,4)/V 5,4 Data:T(5,4)/V 6,1 Data:T(5,4)/V 6,2 Data:T(5,4)/V 6,3 Data:T(5,4)/V 6,4 Data:T(5,4)/V 6,5 Data:T(5,4)/V 6,6 Data:T(5,4)/V 6,7 Data:T(5,4)/V 6,8 Data:T(5,4)/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.