T(7,2)

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T(5,2)

T(4,3)

1 Knot presentations
2 Polynomial invariants
3 "Similar" Knots (within the Atlas)
4 Vassiliev invariants
5 Khovanov Homology
6 Computer Talk
7 Modifying This Page

See other torus knots

Visit T(7,2)'s page at Knotilus!

Visit T(7,2)'s page at the original Knot Atlas!

Edit T(7,2) Quick Notes

Edit T(7,2) Further Notes and Views

[edit] Knot presentations

Planar diagram presentation	X_5,13,6,12 X_13,7,14,6 X_7,1,8,14 X₁₉₂₈ X_9,3,10,2 X_3,11,4,10 X_11,5,12,4
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

[edit] 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

Further Quantum Invariants

T(7,2) Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

In[3]:= K = Knot["T(7,2)"];

In[4]:= Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= $t 3 - t 2 + t -1 + t -1 - t -2 + t -3$

In[5]:= Conway[K][z]

Out[5]= $z 6 + 5 z 4 + 6 z 2 + 1$

In[6]:= Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]= ${1}$

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 7, 6 }

In[8]:= Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]= $- q 10 + q 9 - q 8 + q 7 - q 6 + q 5 + q 3$

In[9]:= HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]= $z 6 a -6 + 6 z 4 a -6 - z 4 a -8 + 10 z 2 a -6 -4 z 2 a -8 + 4 a -6 -3 a -8$

In[10]:= Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]= $z 6 a -6 + z 6 a -8 + z 5 a -7 + z 5 a -9 -6 z 4 a -6 -5 z 4 a -8 + z 4 a -10 -4 z 3 a -7 -3 z 3 a -9 + z 3 a -11 + 10 z 2 a -6 + 7 z 2 a -8 -2 z 2 a -10 + z 2 a -12 + 3 z a -7 + z a -9 - z a -11 + z a -13 -4 a -6 -3 a -8$

[edit] "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {7_1,}

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

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091.

In[3]:= K = Knot["T(7,2)"];

In[4]:= {A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]= { $t 3 - t 2 + t -1 + t -1 - t -2 + t -3$ , $- q 10 + q 9 - q 8 + q 7 - q 6 + q 5 + q 3$ }

In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]= {7_1,}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]= {7_1,}

[edit] Vassiliev invariants

V₂ and V₃:

(6, 14)

[edit] Khovanov Homology

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = 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		\

-1

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 5$	$i = 7$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$
$r = 1$
$r = 2$	${\mathbb Z}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$
$r = 4$	${\mathbb Z}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$
$r = 6$	${\mathbb Z}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$