T(8,3)

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T(15,2).jpg

T(15,2)

T(17,2).jpg

T(17,2)

Contents

T(8,3).jpg See other torus knots

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Edit T(8,3) Further Notes and Views

Banco Internacional do Funchal [1]

Knot presentations

Planar diagram presentation X11,1,12,32 X22,2,23,1 X23,13,24,12 X2,14,3,13 X3,25,4,24 X14,26,15,25 X15,5,16,4 X26,6,27,5 X27,17,28,16 X6,18,7,17 X7,29,8,28 X18,30,19,29 X19,9,20,8 X30,10,31,9 X31,21,32,20 X10,22,11,21
Gauss code 2, -4, -5, 7, 8, -10, -11, 13, 14, -16, -1, 3, 4, -6, -7, 9, 10, -12, -13, 15, 16, -2, -3, 5, 6, -8, -9, 11, 12, -14, -15, 1
Dowker-Thistlethwaite code 22 -24 26 -28 30 -32 2 -4 6 -8 10 -12 14 -16 18 -20
Braid presentation
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BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif

Polynomial invariants

Alexander polynomial t^7-t^6+t^4-t^3+t-1+ t^{-1} - t^{-3} + t^{-4} - t^{-6} + t^{-7}
Conway polynomial z^{14}+13 z^{12}+65 z^{10}+157 z^8+189 z^6+105 z^4+21 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 3, 10 }
Jones polynomial -q^{16}+q^9+q^7
HOMFLY-PT polynomial (db, data sources) z^{14} a^{-14} +14 z^{12} a^{-14} -z^{12} a^{-16} +78 z^{10} a^{-14} -13 z^{10} a^{-16} +221 z^8 a^{-14} -65 z^8 a^{-16} +z^8 a^{-18} +338 z^6 a^{-14} -157 z^6 a^{-16} +8 z^6 a^{-18} +273 z^4 a^{-14} -189 z^4 a^{-16} +21 z^4 a^{-18} +105 z^2 a^{-14} -105 z^2 a^{-16} +21 z^2 a^{-18} +15 a^{-14} -21 a^{-16} +7 a^{-18}
Kauffman polynomial (db, data sources) z^{14} a^{-14} +z^{14} a^{-16} +z^{13} a^{-15} +z^{13} a^{-17} -14 z^{12} a^{-14} -14 z^{12} a^{-16} -13 z^{11} a^{-15} -13 z^{11} a^{-17} +78 z^{10} a^{-14} +78 z^{10} a^{-16} +65 z^9 a^{-15} +65 z^9 a^{-17} -221 z^8 a^{-14} -222 z^8 a^{-16} -z^8 a^{-18} -157 z^7 a^{-15} -157 z^7 a^{-17} +338 z^6 a^{-14} +346 z^6 a^{-16} +8 z^6 a^{-18} +189 z^5 a^{-15} +189 z^5 a^{-17} -273 z^4 a^{-14} -294 z^4 a^{-16} -21 z^4 a^{-18} -105 z^3 a^{-15} -105 z^3 a^{-17} +105 z^2 a^{-14} +126 z^2 a^{-16} +21 z^2 a^{-18} +21 z a^{-15} +21 z a^{-17} -15 a^{-14} -21 a^{-16} -7 a^{-18}
The A2 invariant Data:T(8,3)/QuantumInvariant/A2/1,0
The G2 invariant Data:T(8,3)/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (21, 84)
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(8,3)/V 2,1 Data:T(8,3)/V 3,1 Data:T(8,3)/V 4,1 Data:T(8,3)/V 4,2 Data:T(8,3)/V 4,3 Data:T(8,3)/V 5,1 Data:T(8,3)/V 5,2 Data:T(8,3)/V 5,3 Data:T(8,3)/V 5,4 Data:T(8,3)/V 6,1 Data:T(8,3)/V 6,2 Data:T(8,3)/V 6,3 Data:T(8,3)/V 6,4 Data:T(8,3)/V 6,5 Data:T(8,3)/V 6,6 Data:T(8,3)/V 6,7 Data:T(8,3)/V 6,8 Data:T(8,3)/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=10 is the signature of T(8,3). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
33           1-1
31         1  -1
29         11 0
27       11   0
25     1  1   0
23     11     0
21   11       0
19    1       1
17  1         1
151           1
131           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=9 i=11 i=13 i=15
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}
r=8 {\mathbb Z} {\mathbb Z}
r=9 {\mathbb Z} {\mathbb Z}
r=10 {\mathbb Z}
r=11 {\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

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T(15,2).jpg

T(15,2)

T(17,2).jpg

T(17,2)