T(19,2)

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T(17,2).jpg

T(17,2)

T(10,3).jpg

T(10,3)

T(19,2).jpg See other torus knots

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Knot presentations

Planar diagram presentation X13,33,14,32 X33,15,34,14 X15,35,16,34 X35,17,36,16 X17,37,18,36 X37,19,38,18 X19,1,20,38 X1,21,2,20 X21,3,22,2 X3,23,4,22 X23,5,24,4 X5,25,6,24 X25,7,26,6 X7,27,8,26 X27,9,28,8 X9,29,10,28 X29,11,30,10 X11,31,12,30 X31,13,32,12
Gauss code -8, 9, -10, 11, -12, 13, -14, 15, -16, 17, -18, 19, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 1, -2, 3, -4, 5, -6, 7
Dowker-Thistlethwaite code 20 22 24 26 28 30 32 34 36 38 2 4 6 8 10 12 14 16 18
Braid presentation
BraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gif
BraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gif

Polynomial invariants

Alexander polynomial
Conway polynomial
2nd Alexander ideal (db, data sources)
Determinant and Signature { 19, 18 }
Jones polynomial
HOMFLY-PT polynomial (db, data sources)
Kauffman polynomial (db, data sources)
The A2 invariant Data:T(19,2)/QuantumInvariant/A2/1,0
The G2 invariant Data:T(19,2)/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, ): {}

Vassiliev invariants

V2 and V3: (45, 285)
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(19,2)/V 2,1 Data:T(19,2)/V 3,1 Data:T(19,2)/V 4,1 Data:T(19,2)/V 4,2 Data:T(19,2)/V 4,3 Data:T(19,2)/V 5,1 Data:T(19,2)/V 5,2 Data:T(19,2)/V 5,3 Data:T(19,2)/V 5,4 Data:T(19,2)/V 6,1 Data:T(19,2)/V 6,2 Data:T(19,2)/V 6,3 Data:T(19,2)/V 6,4 Data:T(19,2)/V 6,5 Data:T(19,2)/V 6,6 Data:T(19,2)/V 6,7 Data:T(19,2)/V 6,8 Data:T(19,2)/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 are shown, along with their alternating sums (fixed , alternation over ). The squares with yellow highlighting are those on the "critical diagonals", where or , where 18 is the signature of T(19,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
012345678910111213141516171819χ
57                   1-1
55                    0
53                 11 0
51                    0
49               11   0
47                    0
45             11     0
43                    0
41           11       0
39                    0
37         11         0
35                    0
33       11           0
31                    0
29     11             0
27                    0
25   11               0
23                    0
21  1                 1
191                   1
171                   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

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T(17,2).jpg

T(17,2)

T(10,3).jpg

T(10,3)