# T(3,2)

(Redirected from T(2,3))

### Knot presentations

 Planar diagram presentation X3146 X1524 X5362 Gauss code -2, 3, -1, 2, -3, 1 Dowker-Thistlethwaite code 4 6 2

### Polynomial invariants

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

### "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {3_1,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {3_1,}

### Vassiliev invariants

 V2 and V3: (1, 1)
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(3,2)/V 2,1 Data:T(3,2)/V 3,1 Data:T(3,2)/V 4,1 Data:T(3,2)/V 4,2 Data:T(3,2)/V 4,3 Data:T(3,2)/V 5,1 Data:T(3,2)/V 5,2 Data:T(3,2)/V 5,3 Data:T(3,2)/V 5,4 Data:T(3,2)/V 6,1 Data:T(3,2)/V 6,2 Data:T(3,2)/V 6,3 Data:T(3,2)/V 6,4 Data:T(3,2)/V 6,5 Data:T(3,2)/V 6,6 Data:T(3,2)/V 6,7 Data:T(3,2)/V 6,8 Data:T(3,2)/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=$2 is the signature of T(3,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
0123χ
9   1-1
7    0
5  1 1
31   1
11   1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$ $r=1$ $r=2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.