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KnotPlot

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KnotPlot

PD Presentation:

X1425 X11,14,12,15 X3,13,4,12 X13,3,14,2 X5,20,6,1 X7,18,8,19 X9,16,10,17 X15,10,16,11 X17,8,18,9 X19,6,20,7

Gauss Code:

{-1, 4, -3, 1, -5, 10, -6, 9, -7, 8, -2, 3, -4, 2, -8, 7, -9, 6, -10, 5}

DT (Dowker-Thistlethwaite) Code:

4 12 20 18 16 14 2 10 8 6

Minimum Braid Representative: Length is 13, width is 6 Braid index is 6

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index Reversible 1 1 2 / NotAvailable 1

Alexander Polynomial:

- 4t-1 + 9 - 4t

Conway Polynomial:

1 - 4z2

Other knots with the same Alexander/Conway Polynomial:

{83, ...}

Determinant and Signature:

{17, 0}

Jones Polynomial:

q-8 - q-7 + q-6 - 2q-5 + 2q-4 - 2q-3 + 2q-2 - 2q-1 + 2 - q + q2

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

{...}

A2 (sl(3)) Invariant:

q-26 + q-24 - q-18 - q-16 + q2 + q6 + q8

HOMFLY-PT Polynomial:

a-2 - z2 - a2z2 - a4z2 - a6 - a6z2 + a8

Kauffman Polynomial:

- a-2 + a-2z2 + a-1z3 + z4 - az3 + az5 - 2a2z4 + a2z6 + a3z3 - 3a3z5 + a3z7 + 3a4z4 - 4a4z6 + a4z8 + 4a5z - 11a5z3 + 12a5z5 - 6a5z7 + a5z9 + a6 - 11a6z2 + 21a6z4 - 12a6z6 + 2a6z8 + 4a7z - 14a7z3 + 16a7z5 - 7a7z7 + a7z9 + a8 - 10a8z2 + 15a8z4 - 7a8z6 + a8z8

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

{-4, 6}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=0 is the signature of 101. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 5                     1 j = 3                       j = 1                 2 1   j = -1               1 1     j = -3             1 1       j = -5           1 1         j = -7         1 1           j = -9       1 1             j = -11       1               j = -13   1 1                 j = -15                       j = -17 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-24 - q-23 - q-22 + 2q-21 - q-20 - 2q-19 + 3q-18 - 3q-16 + 3q-15 - 3q-13 + 3q-12 - 3q-10 + 3q-9 - 2q-7 + 3q-6 - q-5 - 2q-4 + 3q-3 - q-2 - 3q-1 + 3 - 2q2 + 2q3 - q5 + q6
3	q-48 - q-47 - q-46 + 2q-44 - 2q-42 - q-41 + 3q-40 + q-39 - 2q-38 - 2q-37 + 2q-36 + 2q-35 - 2q-34 - 2q-33 + 2q-32 + 2q-31 - 2q-30 - 2q-29 + 2q-28 + 2q-27 - 2q-26 - 2q-25 + 2q-24 + 3q-23 - 2q-22 - 3q-21 + q-20 + 4q-19 - 2q-18 - 4q-17 + q-16 + 5q-15 - 2q-14 - 5q-13 + 2q-12 + 6q-11 - 2q-10 - 6q-9 + 2q-8 + 6q-7 - q-6 - 6q-5 + 2q-4 + 4q-3 - 5q-1 + 2 + 2q - 3q3 + 2q4 - 2q7 + 2q8 - q11 + q12

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]:=	PD[Knot[10, 1]]
Out[2]=	PD[X[1, 4, 2, 5], X[11, 14, 12, 15], X[3, 13, 4, 12], X[13, 3, 14, 2], > X[5, 20, 6, 1], X[7, 18, 8, 19], X[9, 16, 10, 17], X[15, 10, 16, 11], > X[17, 8, 18, 9], X[19, 6, 20, 7]]
In[3]:=	GaussCode[Knot[10, 1]]
Out[3]=	GaussCode[-1, 4, -3, 1, -5, 10, -6, 9, -7, 8, -2, 3, -4, 2, -8, 7, -9, 6, -10, > 5]
In[4]:=	DTCode[Knot[10, 1]]
Out[4]=	DTCode[4, 12, 20, 18, 16, 14, 2, 10, 8, 6]
In[5]:=	br = BR[Knot[10, 1]]
Out[5]=	BR[6, {-1, -1, -2, 1, -2, -3, 2, -3, -4, 3, 5, -4, 5}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{6, 13}
In[7]:=	BraidIndex[Knot[10, 1]]
Out[7]=	6
In[8]:=	Show[DrawMorseLink[Knot[10, 1]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 1]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 1, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 1]][t]
Out[10]=	4 9 - - - 4 t t
In[11]:=	Conway[Knot[10, 1]][z]
Out[11]=	2 1 - 4 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[8, 3], Knot[10, 1]}
In[13]:=	{KnotDet[Knot[10, 1]], KnotSignature[Knot[10, 1]]}
Out[13]=	{17, 0}
In[14]:=	Jones[Knot[10, 1]][q]
Out[14]=	-8 -7 -6 2 2 2 2 2 2 2 + q - q + q - -- + -- - -- + -- - - - q + q 5 4 3 2 q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 1]}
In[16]:=	A2Invariant[Knot[10, 1]][q]
Out[16]=	-26 -24 -18 -16 2 6 8 q + q - q - q + q + q + q
In[17]:=	HOMFLYPT[Knot[10, 1]][a, z]
Out[17]=	-2 6 8 2 2 2 4 2 6 2 a - a + a - z - a z - a z - a z
In[18]:=	Kauffman[Knot[10, 1]][a, z]
Out[18]=	2 3 -2 6 8 5 7 z 6 2 8 2 z 3 -a + a + a + 4 a z + 4 a z + -- - 11 a z - 10 a z + -- - a z + 2 a a 3 3 5 3 7 3 4 2 4 4 4 6 4 > a z - 11 a z - 14 a z + z - 2 a z + 3 a z + 21 a z + 8 4 5 3 5 5 5 7 5 2 6 4 6 > 15 a z + a z - 3 a z + 12 a z + 16 a z + a z - 4 a z - 6 6 8 6 3 7 5 7 7 7 4 8 6 8 8 8 > 12 a z - 7 a z + a z - 6 a z - 7 a z + a z + 2 a z + a z + 5 9 7 9 > a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 1]], Vassiliev[3][Knot[10, 1]]}
Out[19]=	{-4, 6}
In[20]:=	Kh[Knot[10, 1]][q, t]
Out[20]=	1 1 1 1 1 1 1 1 1 - + 2 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 17 8 13 7 13 6 11 5 9 5 9 4 7 4 7 3 q t q t q t q t q t q t q t q t 1 1 1 1 1 5 2 > ----- + ----- + ----- + ---- + --- + q t + q t 5 3 5 2 3 2 3 q t q t q t q t q t
In[21]:=	ColouredJones[Knot[10, 1], 2][q]
Out[21]=	-24 -23 -22 2 -20 2 3 3 3 3 3 3 3 + q - q - q + --- - q - --- + --- - --- + --- - --- + --- - --- + 21 19 18 16 15 13 12 10 q q q q q q q q 3 2 3 -5 2 3 -2 3 2 3 5 6 > -- - -- + -- - q - -- + -- - q - - - 2 q + 2 q - q + q 9 7 6 4 3 q q q q q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 101

949 102