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T(7,2).jpg See other torus knots

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Edit T(7,2) Quick Notes

See also 7_1.

Edit T(7,2) Further Notes and Views

Knot presentations

Planar diagram presentation X5,13,6,12 X13,7,14,6 X7,1,8,14 X1928 X9,3,10,2 X3,11,4,10 X11,5,12,4
Gauss code -4, 5, -6, 7, -1, 2, -3, 4, -5, 6, -7, 1, -2, 3
Dowker-Thistlethwaite code 8 10 12 14 2 4 6
Braid presentation

Polynomial invariants

Alexander polynomial t^3-t^2+t-1+ t^{-1} - t^{-2} + t^{-3}
Conway polynomial z^6+5 z^4+6 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 7, 6 }
Jones polynomial -q^{10}+q^9-q^8+q^7-q^6+q^5+q^3
HOMFLY-PT polynomial (db, data sources) z^6 a^{-6} +6 z^4 a^{-6} -z^4 a^{-8} +10 z^2 a^{-6} -4 z^2 a^{-8} +4 a^{-6} -3 a^{-8}
Kauffman polynomial (db, data sources) z^6 a^{-6} +z^6 a^{-8} +z^5 a^{-7} +z^5 a^{-9} -6 z^4 a^{-6} -5 z^4 a^{-8} +z^4 a^{-10} -4 z^3 a^{-7} -3 z^3 a^{-9} +z^3 a^{-11} +10 z^2 a^{-6} +7 z^2 a^{-8} -2 z^2 a^{-10} +z^2 a^{-12} +3 z a^{-7} +z a^{-9} -z a^{-11} +z a^{-13} -4 a^{-6} -3 a^{-8}
The A2 invariant Data:T(7,2)/QuantumInvariant/A2/1,0
The G2 invariant Data:T(7,2)/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {7_1,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {7_1,}

Vassiliev invariants

V2 and V3: (6, 14)
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(7,2)/V 2,1 Data:T(7,2)/V 3,1 Data:T(7,2)/V 4,1 Data:T(7,2)/V 4,2 Data:T(7,2)/V 4,3 Data:T(7,2)/V 5,1 Data:T(7,2)/V 5,2 Data:T(7,2)/V 5,3 Data:T(7,2)/V 5,4 Data:T(7,2)/V 6,1 Data:T(7,2)/V 6,2 Data:T(7,2)/V 6,3 Data:T(7,2)/V 6,4 Data:T(7,2)/V 6,5 Data:T(7,2)/V 6,6 Data:T(7,2)/V 6,7 Data:T(7,2)/V 6,8 Data:T(7,2)/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=6 is the signature of T(7,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
21       1-1
19        0
17     11 0
15        0
13   11   0
11        0
9  1     1
71       1
51       1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=5 i=7
r=0 {\mathbb Z} {\mathbb Z}
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}

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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