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

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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
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
Braid presentation

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 5, 4 }
Jones polynomial
HOMFLY-PT polynomial (db, data sources)
Kauffman polynomial (db, data sources)
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, ): {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 are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where 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)


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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