10 57
From Knot Atlas
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See the full Rolfsen Knot Table. Visit 10 57's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) |
Knot presentations
Planar diagram presentation | X1425 X3849 X9,15,10,14 X5,13,6,12 X13,7,14,6 X11,19,12,18 X15,1,16,20 X19,17,20,16 X17,11,18,10 X7283 |
Gauss code | -1, 10, -2, 1, -4, 5, -10, 2, -3, 9, -6, 4, -5, 3, -7, 8, -9, 6, -8, 7 |
Dowker-Thistlethwaite code | 4 8 12 2 14 18 6 20 10 16 |
Conway Notation | [221,21,2] |
Minimum Braid Representative | A Morse Link Presentation | An Arc Presentation | ||||
Length is 11, width is 4, Braid index is 4 |
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![]() [{12, 2}, {1, 10}, {9, 11}, {10, 12}, {11, 14}, {3, 13}, {2, 9}, {8, 4}, {7, 3}, {5, 8}, {14, 7}, {4, 6}, {13, 5}, {6, 1}] |
[edit Notes on presentations of 10 57]
Three dimensional invariants
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Four dimensional invariants
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Polynomial invariants
"Similar" Knots (within the Atlas)
Same Alexander/Conway Polynomial: {K11n40, K11n46,}
Same Jones Polynomial (up to mirroring, ): {}
Vassiliev invariants
V2 and V3: | (4, 6) |
V2,1 through V6,9: |
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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 2 is the signature of 10 57. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. |
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Integral Khovanov Homology
(db, data source) |
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