T(9,2)

From Knot Atlas
Jump to: navigation, search

T(4,3).jpg

T(4,3)

T(5,3).jpg

T(5,3)

Contents

T(9,2).jpg See other torus knots

Visit T(9,2) at Knotilus!

Edit T(9,2) Quick Notes

See also 9_1.

Edit T(9,2) Further Notes and Views


Knot presentations

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

Polynomial invariants

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_1,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {9_1,}

Vassiliev invariants

V2 and V3: (10, 30)
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(9,2)/V 2,1 Data:T(9,2)/V 3,1 Data:T(9,2)/V 4,1 Data:T(9,2)/V 4,2 Data:T(9,2)/V 4,3 Data:T(9,2)/V 5,1 Data:T(9,2)/V 5,2 Data:T(9,2)/V 5,3 Data:T(9,2)/V 5,4 Data:T(9,2)/V 6,1 Data:T(9,2)/V 6,2 Data:T(9,2)/V 6,3 Data:T(9,2)/V 6,4 Data:T(9,2)/V 6,5 Data:T(9,2)/V 6,6 Data:T(9,2)/V 6,7 Data:T(9,2)/V 6,8 Data:T(9,2)/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=8 is the signature of T(9,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
0123456789χ
27         1-1
25          0
23       11 0
21          0
19     11   0
17          0
15   11     0
13          0
11  1       1
91         1
71         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=7 i=9
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}

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).

Back to the top.

T(4,3).jpg

T(4,3)

T(5,3).jpg

T(5,3)