T(11,2)

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T(5,3).jpg

T(5,3)

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T(13,2)

Contents

T(11,2).jpg See other torus knots

Visit T(11,2) at Knotilus!

Edit T(11,2) Quick Notes

See also K11a367.

Edit T(11,2) Further Notes and Views


Knot presentations

Planar diagram presentation X5,17,6,16 X17,7,18,6 X7,19,8,18 X19,9,20,8 X9,21,10,20 X21,11,22,10 X11,1,12,22 X1,13,2,12 X13,3,14,2 X3,15,4,14 X15,5,16,4
Gauss code -8, 9, -10, 11, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 1, -2, 3, -4, 5, -6, 7
Dowker-Thistlethwaite code 12 14 16 18 20 22 2 4 6 8 10
Braid presentation
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gif

Polynomial invariants

Alexander polynomial t^5-t^4+t^3-t^2+t-1+ t^{-1} - t^{-2} + t^{-3} - t^{-4} + t^{-5}
Conway polynomial z^{10}+9 z^8+28 z^6+35 z^4+15 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 11, 10 }
Jones polynomial -q^{16}+q^{15}-q^{14}+q^{13}-q^{12}+q^{11}-q^{10}+q^9-q^8+q^7+q^5
HOMFLY-PT polynomial (db, data sources) z^{10} a^{-10} +10 z^8 a^{-10} -z^8 a^{-12} +36 z^6 a^{-10} -8 z^6 a^{-12} +56 z^4 a^{-10} -21 z^4 a^{-12} +35 z^2 a^{-10} -20 z^2 a^{-12} +6 a^{-10} -5 a^{-12}
Kauffman polynomial (db, data sources) z^{10} a^{-10} +z^{10} a^{-12} +z^9 a^{-11} +z^9 a^{-13} -10 z^8 a^{-10} -9 z^8 a^{-12} +z^8 a^{-14} -8 z^7 a^{-11} -7 z^7 a^{-13} +z^7 a^{-15} +36 z^6 a^{-10} +29 z^6 a^{-12} -6 z^6 a^{-14} +z^6 a^{-16} +21 z^5 a^{-11} +15 z^5 a^{-13} -5 z^5 a^{-15} +z^5 a^{-17} -56 z^4 a^{-10} -41 z^4 a^{-12} +10 z^4 a^{-14} -4 z^4 a^{-16} +z^4 a^{-18} -20 z^3 a^{-11} -10 z^3 a^{-13} +6 z^3 a^{-15} -3 z^3 a^{-17} +z^3 a^{-19} +35 z^2 a^{-10} +25 z^2 a^{-12} -4 z^2 a^{-14} +3 z^2 a^{-16} -2 z^2 a^{-18} +z^2 a^{-20} +5 z a^{-11} +z a^{-13} -z a^{-15} +z a^{-17} -z a^{-19} +z a^{-21} -6 a^{-10} -5 a^{-12}
The A2 invariant Data:T(11,2)/QuantumInvariant/A2/1,0
The G2 invariant Data:T(11,2)/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a367,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a367,}

Vassiliev invariants

V2 and V3: (15, 55)
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(11,2)/V 2,1 Data:T(11,2)/V 3,1 Data:T(11,2)/V 4,1 Data:T(11,2)/V 4,2 Data:T(11,2)/V 4,3 Data:T(11,2)/V 5,1 Data:T(11,2)/V 5,2 Data:T(11,2)/V 5,3 Data:T(11,2)/V 5,4 Data:T(11,2)/V 6,1 Data:T(11,2)/V 6,2 Data:T(11,2)/V 6,3 Data:T(11,2)/V 6,4 Data:T(11,2)/V 6,5 Data:T(11,2)/V 6,6 Data:T(11,2)/V 6,7 Data:T(11,2)/V 6,8 Data:T(11,2)/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=10 is the signature of T(11,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
33           1-1
31            0
29         11 0
27            0
25       11   0
23            0
21     11     0
19            0
17   11       0
15            0
13  1         1
111           1
91           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=9 i=11
r=0 {\mathbb Z} {\mathbb Z}
r=1
r=2 {\mathbb Z}
r=3 {\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}
r=5 {\mathbb Z}_2 {\mathbb Z}
r=6 {\mathbb Z}
r=7 {\mathbb Z}_2 {\mathbb Z}
r=8 {\mathbb Z}
r=9 {\mathbb Z}_2 {\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(5,3).jpg

T(5,3)

T(13,2).jpg

T(13,2)