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T(5,2).jpg See other torus knots

Visit T(5,2) at Knotilus!

Edit T(5,2) Quick Notes

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).

Edit T(5,2) Further Notes and Views

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

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
Braid presentation

Polynomial invariants

Alexander polynomial t^2-t+1- t^{-1} + t^{-2}
Conway polynomial z^4+3 z^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} +4 z^2 a^{-4} -z^2 a^{-6} +3 a^{-4} -2 a^{-6}
Kauffman polynomial (db, data sources) z^4 a^{-4} +z^4 a^{-6} +z^3 a^{-5} +z^3 a^{-7} -4 z^2 a^{-4} -3 z^2 a^{-6} +z^2 a^{-8} -2 z a^{-5} -z a^{-7} +z a^{-9} +3 a^{-4} +2 a^{-6}
The A2 invariant Data:T(5,2)/QuantumInvariant/A2/1,0
The G2 invariant Data:T(5,2)/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {5_1, 10_132,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {5_1, 10_132,}

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

V2,1 through V6,9 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^rq^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 \
15     1-1
13      0
11   11 0
9      0
7  1   1
51     1
31     1
Integral Khovanov Homology

(db, data source)

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

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Torus Knot Page master template (intermediate).

See/edit the Torus Knot_Splice_Base (expert).

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