T(3,2)

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T(3,2).jpg

T(3,2)

T(5,2).jpg

T(5,2)

Contents

T(3,2).jpg See other torus knots

Visit T(3,2) at Knotilus!

Edit T(3,2) Quick Notes

See 3_1.

Edit T(3,2) Further Notes and Views


Knot presentations

Planar diagram presentation X3146 X1524 X5362
Gauss code -2, 3, -1, 2, -3, 1
Dowker-Thistlethwaite code 4 6 2
Braid presentation
BraidPart1.gifBraidPart1.gifBraidPart1.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gif

Polynomial invariants

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {3_1,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {3_1,}

Vassiliev invariants

V2 and V3: (1, 1)
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(3,2)/V 2,1 Data:T(3,2)/V 3,1 Data:T(3,2)/V 4,1 Data:T(3,2)/V 4,2 Data:T(3,2)/V 4,3 Data:T(3,2)/V 5,1 Data:T(3,2)/V 5,2 Data:T(3,2)/V 5,3 Data:T(3,2)/V 5,4 Data:T(3,2)/V 6,1 Data:T(3,2)/V 6,2 Data:T(3,2)/V 6,3 Data:T(3,2)/V 6,4 Data:T(3,2)/V 6,5 Data:T(3,2)/V 6,6 Data:T(3,2)/V 6,7 Data:T(3,2)/V 6,8 Data:T(3,2)/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=2 is the signature of T(3,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
0123χ
9   1-1
7    0
5  1 1
31   1
11   1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=0 {\mathbb Z} {\mathbb Z}
r=1
r=2 {\mathbb Z}
r=3 {\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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T(3,2).jpg

T(3,2)

T(5,2).jpg

T(5,2)