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934 936

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KnotPlot

PD Presentation:

X1829 X7,14,8,15 X5,16,6,17 X9,18,10,1 X15,6,16,7 X17,10,18,11 X13,2,14,3 X3,12,4,13 X11,4,12,5

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

8 12 16 14 18 4 2 6 10

Minimum Braid Representative: Length is 14, width is 5 Braid index is 5

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index Reversible 2--3 1 3 / 4--6 2

Alexander Polynomial:

7t-1 - 13 + 7t

Conway Polynomial:

1 + 7z2

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{27, -2}

Jones Polynomial:

- q-10 + q-9 - 3q-8 + 4q-7 - 3q-6 + 5q-5 - 4q-4 + 3q-3 - 2q-2 + q-1

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

{...}

A2 (sl(3)) Invariant:

- q-32 - q-30 - 2q-26 - q-24 + q-22 + q-20 + 3q-18 + 2q-16 + q-14 - q-10 + q-8 - q-4 + q-2

HOMFLY-PT Polynomial:

a2z2 + 2a4z2 + 3a6 + 3a6z2 - a8 + a8z2 - a10

Kauffman Polynomial:

a2z2 + 2a3z3 - 2a4z2 + 3a4z4 - 6a5z3 + 4a5z5 - 3a6 + 12a6z2 - 15a6z4 + 5a6z6 - a7z + 3a7z3 - 8a7z5 + 3a7z7 - a8 + 16a8z2 - 15a8z4 + a8z6 + a8z8 - 9a9z + 23a9z3 - 18a9z5 + 4a9z7 + a10 + a10z2 + 3a10z4 - 4a10z6 + a10z8 - 8a11z + 12a11z3 - 6a11z5 + a11z7

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

{7, -18}

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=-2 is the signature of 935. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -1                   1 j = -3                 2 1 j = -5               1     j = -7             3 2     j = -9           2 1       j = -11         1 3         j = -13       3 2           j = -15       1             j = -17   1 3               j = -19                     j = -21 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-29 - q-28 - q-27 + 4q-26 - q-25 - 6q-24 + 7q-23 - 11q-21 + 8q-20 + 4q-19 - 13q-18 + 7q-17 + 9q-16 - 13q-15 + 4q-14 + 9q-13 - 10q-12 + 8q-10 - 7q-9 + 5q-7 - 4q-6 + 2q-5 + q-4 - 2q-3 + q-2

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[9, 35]]
Out[2]=	PD[X[1, 8, 2, 9], X[7, 14, 8, 15], X[5, 16, 6, 17], X[9, 18, 10, 1], > X[15, 6, 16, 7], X[17, 10, 18, 11], X[13, 2, 14, 3], X[3, 12, 4, 13], > X[11, 4, 12, 5]]
In[3]:=	GaussCode[Knot[9, 35]]
Out[3]=	GaussCode[-1, 7, -8, 9, -3, 5, -2, 1, -4, 6, -9, 8, -7, 2, -5, 3, -6, 4]
In[4]:=	DTCode[Knot[9, 35]]
Out[4]=	DTCode[8, 12, 16, 14, 18, 4, 2, 6, 10]
In[5]:=	br = BR[Knot[9, 35]]
Out[5]=	BR[5, {-1, -1, -2, 1, -2, -2, -3, 2, 2, -4, 3, -2, -4, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 14}
In[7]:=	BraidIndex[Knot[9, 35]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[9, 35]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[9, 35]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, {2, 3}, 1, 3, {4, 6}, 2}
In[10]:=	alex = Alexander[Knot[9, 35]][t]
Out[10]=	7 -13 + - + 7 t t
In[11]:=	Conway[Knot[9, 35]][z]
Out[11]=	2 1 + 7 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 35]}
In[13]:=	{KnotDet[Knot[9, 35]], KnotSignature[Knot[9, 35]]}
Out[13]=	{27, -2}
In[14]:=	Jones[Knot[9, 35]][q]
Out[14]=	-10 -9 3 4 3 5 4 3 2 1 -q + q - -- + -- - -- + -- - -- + -- - -- + - 8 7 6 5 4 3 2 q q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 35]}
In[16]:=	A2Invariant[Knot[9, 35]][q]
Out[16]=	-32 -30 2 -24 -22 -20 3 2 -14 -10 -8 -4 -q - q - --- - q + q + q + --- + --- + q - q + q - q + 26 18 16 q q q -2 > q
In[17]:=	HOMFLYPT[Knot[9, 35]][a, z]
Out[17]=	6 8 10 2 2 4 2 6 2 8 2 3 a - a - a + a z + 2 a z + 3 a z + a z
In[18]:=	Kauffman[Knot[9, 35]][a, z]
Out[18]=	6 8 10 7 9 11 2 2 4 2 6 2 -3 a - a + a - a z - 9 a z - 8 a z + a z - 2 a z + 12 a z + 8 2 10 2 3 3 5 3 7 3 9 3 11 3 > 16 a z + a z + 2 a z - 6 a z + 3 a z + 23 a z + 12 a z + 4 4 6 4 8 4 10 4 5 5 7 5 9 5 > 3 a z - 15 a z - 15 a z + 3 a z + 4 a z - 8 a z - 18 a z - 11 5 6 6 8 6 10 6 7 7 9 7 11 7 > 6 a z + 5 a z + a z - 4 a z + 3 a z + 4 a z + a z + 8 8 10 8 > a z + a z
In[19]:=	{Vassiliev[2][Knot[9, 35]], Vassiliev[3][Knot[9, 35]]}
Out[19]=	{7, -18}
In[20]:=	Kh[Knot[9, 35]][q, t]
Out[20]=	-3 1 1 1 3 1 3 2 1 q + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + q 21 9 17 8 17 7 15 6 13 6 13 5 11 5 q t q t q t q t q t q t q t 3 2 1 3 2 1 2 > ------ + ----- + ----- + ----- + ----- + ----- + ---- 11 4 9 4 9 3 7 3 7 2 5 2 3 q t q t q t q t q t q t q t
In[21]:=	ColouredJones[Knot[9, 35], 2][q]
Out[21]=	-29 -28 -27 4 -25 6 7 11 8 4 13 7 q - q - q + --- - q - --- + --- - --- + --- + --- - --- + --- + 26 24 23 21 20 19 18 17 q q q q q q q q 9 13 4 9 10 8 7 5 4 2 -4 2 -2 > --- - --- + --- + --- - --- + --- - -- + -- - -- + -- + q - -- + q 16 15 14 13 12 10 9 7 6 5 3 q q q q q q q q q q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 935

934 936