T(5,3)

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T(9,2).jpg

T(9,2)

T(11,2).jpg

T(11,2)

Contents

T(5,3).jpg See other torus knots

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Knot presentations

Planar diagram presentation X7,1,8,20 X14,2,15,1 X15,9,16,8 X2,10,3,9 X3,17,4,16 X10,18,11,17 X11,5,12,4 X18,6,19,5 X19,13,20,12 X6,14,7,13
Gauss code 2, -4, -5, 7, 8, -10, -1, 3, 4, -6, -7, 9, 10, -2, -3, 5, 6, -8, -9, 1
Dowker-Thistlethwaite code 14 -16 18 -20 2 -4 6 -8 10 -12
Braid presentation
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif

Polynomial invariants

Alexander polynomial t^4-t^3+t-1+ t^{-1} - t^{-3} + t^{-4}
Conway polynomial z^8+7 z^6+14 z^4+8 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 1, 8 }
Jones polynomial -q^{10}+q^6+q^4
HOMFLY-PT polynomial (db, data sources) z^8 a^{-8} +8 z^6 a^{-8} -z^6 a^{-10} +21 z^4 a^{-8} -7 z^4 a^{-10} +21 z^2 a^{-8} -14 z^2 a^{-10} +z^2 a^{-12} +7 a^{-8} -8 a^{-10} +2 a^{-12}
Kauffman polynomial (db, data sources) z^8 a^{-8} +z^8 a^{-10} +z^7 a^{-9} +z^7 a^{-11} -8 z^6 a^{-8} -8 z^6 a^{-10} -7 z^5 a^{-9} -7 z^5 a^{-11} +21 z^4 a^{-8} +21 z^4 a^{-10} +14 z^3 a^{-9} +14 z^3 a^{-11} -21 z^2 a^{-8} -22 z^2 a^{-10} -z^2 a^{-12} -8 z a^{-9} -8 z a^{-11} +7 a^{-8} +8 a^{-10} +2 a^{-12}
The A2 invariant Data:T(5,3)/QuantumInvariant/A2/1,0
The G2 invariant Data:T(5,3)/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_124,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {10_124,}

Vassiliev invariants

V2 and V3: (8, 20)
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(5,3)/V 2,1 Data:T(5,3)/V 3,1 Data:T(5,3)/V 4,1 Data:T(5,3)/V 4,2 Data:T(5,3)/V 4,3 Data:T(5,3)/V 5,1 Data:T(5,3)/V 5,2 Data:T(5,3)/V 5,3 Data:T(5,3)/V 5,4 Data:T(5,3)/V 6,1 Data:T(5,3)/V 6,2 Data:T(5,3)/V 6,3 Data:T(5,3)/V 6,4 Data:T(5,3)/V 6,5 Data:T(5,3)/V 6,6 Data:T(5,3)/V 6,7 Data:T(5,3)/V 6,8 Data:T(5,3)/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=8 is the signature of T(5,3). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567χ
21       1-1
19     1  -1
17     11 0
15   11   0
13    1   1
11  1     1
91       1
71       1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=5 i=7 i=9
r=0 {\mathbb Z} {\mathbb Z}
r=1
r=2 {\mathbb Z}
r=3 {\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z} {\mathbb Z}
r=5 {\mathbb Z} {\mathbb Z}
r=6 {\mathbb Z}
r=7 {\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

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See/edit the Torus Knot_Splice_Base (expert).

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T(9,2).jpg

T(9,2)

T(11,2).jpg

T(11,2)