From Knot Atlas
Jump to: navigation, search






T(4,3).jpg See other torus knots

Visit T(4,3) at Knotilus!

Edit T(4,3) Quick Notes

See also 8_19.

Edit T(4,3) Further Notes and Views

Knot presentations

Planar diagram presentation X5,11,6,10 X16,12,1,11 X1726 X12,8,13,7 X13,3,14,2 X8493 X9,15,10,14 X4,16,5,15
Gauss code -3, 5, 6, -8, -1, 3, 4, -6, -7, 1, 2, -4, -5, 7, 8, -2
Dowker-Thistlethwaite code 6 -8 10 -12 14 -16 2 -4
Braid presentation

Polynomial invariants

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_19,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {8_19,}

Vassiliev invariants

V2 and V3: (5, 10)
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(4,3)/V 2,1 Data:T(4,3)/V 3,1 Data:T(4,3)/V 4,1 Data:T(4,3)/V 4,2 Data:T(4,3)/V 4,3 Data:T(4,3)/V 5,1 Data:T(4,3)/V 5,2 Data:T(4,3)/V 5,3 Data:T(4,3)/V 5,4 Data:T(4,3)/V 6,1 Data:T(4,3)/V 6,2 Data:T(4,3)/V 6,3 Data:T(4,3)/V 6,4 Data:T(4,3)/V 6,5 Data:T(4,3)/V 6,6 Data:T(4,3)/V 6,7 Data:T(4,3)/V 6,8 Data:T(4,3)/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 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(4,3). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
17     1-1
15     1-1
13   11 0
11    1 1
9  1   1
71     1
51     1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 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} {\mathbb Z}
r=5 {\mathbb Z} {\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.