T(8,3)

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T(15,2).jpg

T(15,2)

T(17,2).jpg

T(17,2)

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

Visit T(8,3) at Knotilus!

Edit T(8,3) Quick Notes


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
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 3, 10 }
Jones polynomial
HOMFLY-PT polynomial (db, data sources)
Kauffman polynomial (db, data sources)
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, ): {}

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 are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where 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)

  

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(15,2).jpg

T(15,2)

T(17,2).jpg

T(17,2)