© | Dror Bar-Natan: The Knot Atlas: Torus Knots:

T(7,3) T(15,2)

TubePlot

This page is passe. Go here instead!    The 15-Crossing Torus Knot T(5,4) Visit T(5,4)'s page at Knotilus! Acknowledgement

PD Presentation:

X17,25,18,24 X10,26,11,25 X3,27,4,26 X11,19,12,18 X4,20,5,19 X27,21,28,20 X5,13,6,12 X28,14,29,13 X21,15,22,14 X29,7,30,6 X22,8,23,7 X15,9,16,8 X23,1,24,30 X16,2,17,1 X9,3,10,2

Gauss Code:

{14, 15, -3, -5, -7, 10, 11, 12, -15, -2, -4, 7, 8, 9, -12, -14, -1, 4, 5, 6, -9, -11, -13, 1, 2, 3, -6, -8, -10, 13}

Braid Representative:

Alexander Polynomial:

t-6 - t-5 + t-2 - 1 + t2 - t5 + t6

Conway Polynomial:

1 + 15z2 + 56z4 + 77z6 + 44z8 + 11z10 + z12

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{5, 8}

Jones Polynomial:

q6 + q8 + q10 - q11 - q13

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

Coloured Jones Polynomial (in the 3-dimensional representation of sl(2); n=2):

q12 + q15 + q18 - q23 - q26 - q29 + q30 - q32 + q33 - q35 + q36 - q38 + q39

A2 (sl(3)) Invariant:

q22 + q24 + 2q26 + 2q28 + 3q30 + 2q32 + q34 - q36 - 2q38 - 3q40 - 3q42 - 2q44 - q46 + q48 + q50 + q52

Kauffman Polynomial:

- a-19z + a-18 - a-18z2 - 8a-17z + 14a-17z3 - 7a-17z5 + a-17z7 + 9a-16 - 22a-16z2 + 21a-16z4 - 8a-16z6 + a-16z8 - 28a-15z + 84a-15z3 - 91a-15z5 + 46a-15z7 - 11a-15z9 + a-15z11 + 21a-14 - 91a-14z2 + 154a-14z4 - 129a-14z6 + 56a-14z8 - 12a-14z10 + a-14z12 - 21a-13z + 70a-13z3 - 84a-13z5 + 45a-13z7 - 11a-13z9 + a-13z11 + 14a-12 - 70a-12z2 + 133a-12z4 - 121a-12z6 + 55a-12z8 - 12a-12z10 + a-12z12

V2 and V3, the type 2 and 3 Vassiliev invariants:

{15, 50}

Khovanov Homology. The coefficients of the monomials t^rq^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-2r=s+1 or j-2r=s+1, where s=8 is the signature of T(5,4). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	TubePlot[TorusKnot[5, 4]]
Out[2]=	-Graphics-
In[3]:=	Crossings[TorusKnot[5, 4]]
Out[3]=	15
In[4]:=	PD[TorusKnot[5, 4]]
Out[4]=	PD[X[17, 25, 18, 24], X[10, 26, 11, 25], X[3, 27, 4, 26], X[11, 19, 12, 18], > X[4, 20, 5, 19], X[27, 21, 28, 20], X[5, 13, 6, 12], X[28, 14, 29, 13], > X[21, 15, 22, 14], X[29, 7, 30, 6], X[22, 8, 23, 7], X[15, 9, 16, 8], > X[23, 1, 24, 30], X[16, 2, 17, 1], X[9, 3, 10, 2]]
In[5]:=	GaussCode[TorusKnot[5, 4]]
Out[5]=	GaussCode[14, 15, -3, -5, -7, 10, 11, 12, -15, -2, -4, 7, 8, 9, -12, -14, -1, > 4, 5, 6, -9, -11, -13, 1, 2, 3, -6, -8, -10, 13]
In[6]:=	BR[TorusKnot[5, 4]]
Out[6]=	BR[4, {1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3}]
In[7]:=	alex = Alexander[TorusKnot[5, 4]][t]
Out[7]=	-6 -5 -2 2 5 6 -1 + t - t + t + t - t + t
In[8]:=	Conway[TorusKnot[5, 4]][z]
Out[8]=	2 4 6 8 10 12 1 + 15 z + 56 z + 77 z + 44 z + 11 z + z
In[9]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[9]=	{}
In[10]:=	{KnotDet[TorusKnot[5, 4]], KnotSignature[TorusKnot[5, 4]]}
Out[10]=	{5, 8}
In[11]:=	J=Jones[TorusKnot[5, 4]][q]
Out[11]=	6 8 10 11 13 q + q + q - q - q
In[12]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[12]=	{}
In[13]:=	ColouredJones[TorusKnot[5, 4], 2][q]
Out[13]=	12 15 18 23 26 29 30 32 33 35 36 38 39 q + q + q - q - q - q + q - q + q - q + q - q + q
In[14]:=	A2Invariant[TorusKnot[5, 4]][q]
Out[14]=	22 24 26 28 30 32 34 36 38 40 42 q + q + 2 q + 2 q + 3 q + 2 q + q - q - 2 q - 3 q - 3 q - 44 46 48 50 52 > 2 q - q + q + q + q
In[15]:=	Kauffman[TorusKnot[5, 4]][a, z]
Out[15]=	2 2 2 -18 9 21 14 z 8 z 28 z 21 z z 22 z 91 z a + --- + --- + --- - --- - --- - ---- - ---- - --- - ----- - ----- - 16 14 12 19 17 15 13 18 16 14 a a a a a a a a a a 2 3 3 3 4 4 4 5 5 70 z 14 z 84 z 70 z 21 z 154 z 133 z 7 z 91 z > ----- + ----- + ----- + ----- + ----- + ------ + ------ - ---- - ----- - 12 17 15 13 16 14 12 17 15 a a a a a a a a a 5 6 6 6 7 7 7 8 8 84 z 8 z 129 z 121 z z 46 z 45 z z 56 z > ----- - ---- - ------ - ------ + --- + ----- + ----- + --- + ----- + 13 16 14 12 17 15 13 16 14 a a a a a a a a a 8 9 9 10 10 11 11 12 12 55 z 11 z 11 z 12 z 12 z z z z z > ----- - ----- - ----- - ------ - ------ + --- + --- + --- + --- 12 15 13 14 12 15 13 14 12 a a a a a a a a a
In[16]:=	{Vassiliev[2][TorusKnot[5, 4]], Vassiliev[3][TorusKnot[5, 4]]}
Out[16]=	{15, 50}
In[17]:=	Kh[TorusKnot[5, 4]][q, t]
Out[17]=	11 13 15 2 19 3 17 4 19 4 21 5 23 5 19 6 q + q + q t + q t + q t + q t + q t + q t + q t + 21 6 23 7 25 7 23 8 27 9 > q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: Torus Knots: The Torus Knot T(5,4)

T(7,3) T(15,2)