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1032 1034

KnotPlot

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KnotPlot

PD Presentation:

X6271 X14,6,15,5 X20,15,1,16 X16,7,17,8 X8,19,9,20 X18,9,19,10 X10,17,11,18 X2,14,3,13 X12,4,13,3 X4,12,5,11

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 12 14 16 18 4 2 20 10 8

Minimum Braid Representative: Length is 12, 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 FullyAmphicheiral 1 2 2 / NotAvailable 1

Alexander Polynomial:

4t-2 - 16t-1 + 25 - 16t + 4t2

Conway Polynomial:

1 + 4z4

Other knots with the same Alexander/Conway Polynomial:

{K11a333, ...}

Determinant and Signature:

{65, 0}

Jones Polynomial:

- q-5 + 3q-4 - 5q-3 + 8q-2 - 10q-1 + 11 - 10q + 8q2 - 5q3 + 3q4 - q5

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

{...}

A2 (sl(3)) Invariant:

- q-16 + q-14 + q-12 - 2q-10 + 2q-8 - q-4 + 2q-2 - 1 + 2q2 - q4 + 2q8 - 2q10 + q12 + q14 - q16

HOMFLY-PT Polynomial:

- a-4z2 + a-2z4 + 1 + 2z2 + 2z4 + a2z4 - a4z2

Kauffman Polynomial:

- 2a-5z3 + a-5z5 + 3a-4z2 - 7a-4z4 + 3a-4z6 - 2a-3z + 6a-3z3 - 9a-3z5 + 4a-3z7 + a-2z4 - 4a-2z6 + 3a-2z8 - 6a-1z + 18a-1z3 - 16a-1z5 + 5a-1z7 + a-1z9 + 1 - 6z2 + 16z4 - 14z6 + 6z8 - 6az + 18az3 - 16az5 + 5az7 + az9 + a2z4 - 4a2z6 + 3a2z8 - 2a3z + 6a3z3 - 9a3z5 + 4a3z7 + 3a4z2 - 7a4z4 + 3a4z6 - 2a5z3 + a5z5

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

{0, 0}

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 1033. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                     1 j = 9                   2   j = 7                 3 1   j = 5               5 2     j = 3             5 3       j = 1           6 5         j = -1         5 6           j = -3       3 5             j = -5     2 5               j = -7   1 3                 j = -9   2                   j = -11 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-15 - 3q-14 + q-13 + 8q-12 - 14q-11 + 27q-9 - 31q-8 - 9q-7 + 58q-6 - 48q-5 - 28q-4 + 89q-3 - 55q-2 - 47q-1 + 103 - 47q - 55q2 + 89q3 - 28q4 - 48q5 + 58q6 - 9q7 - 31q8 + 27q9 - 14q11 + 8q12 + q13 - 3q14 + q15
3	- q-30 + 3q-29 - q-28 - 4q-27 - q-26 + 11q-25 + 2q-24 - 21q-23 - 6q-22 + 35q-21 + 15q-20 - 54q-19 - 31q-18 + 74q-17 + 62q-16 - 99q-15 - 98q-14 + 110q-13 + 156q-12 - 123q-11 - 209q-10 + 113q-9 + 275q-8 - 103q-7 - 327q-6 + 77q-5 + 373q-4 - 50q-3 - 399q-2 + 15q-1 + 413 + 15q - 399q2 - 50q3 + 373q4 + 77q5 - 327q6 - 103q7 + 275q8 + 113q9 - 209q10 - 123q11 + 156q12 + 110q13 - 98q14 - 99q15 + 62q16 + 74q17 - 31q18 - 54q19 + 15q20 + 35q21 - 6q22 - 21q23 + 2q24 + 11q25 - q26 - 4q27 - q28 + 3q29 - q30
4	q-50 - 3q-49 + q-48 + 4q-47 - 3q-46 + 4q-45 - 14q-44 + 6q-43 + 18q-42 - 13q-41 + 12q-40 - 47q-39 + 15q-38 + 61q-37 - 25q-36 + 20q-35 - 129q-34 + 19q-33 + 153q-32 - 2q-31 + 50q-30 - 302q-29 - 50q-28 + 273q-27 + 127q-26 + 197q-25 - 548q-24 - 286q-23 + 292q-22 + 351q-21 + 584q-20 - 725q-19 - 681q-18 + 73q-17 + 529q-16 + 1179q-15 - 689q-14 - 1071q-13 - 356q-12 + 531q-11 + 1785q-10 - 459q-9 - 1299q-8 - 814q-7 + 369q-6 + 2200q-5 - 159q-4 - 1320q-3 - 1155q-2 + 124q-1 + 2345 + 124q - 1155q2 - 1320q3 - 159q4 + 2200q5 + 369q6 - 814q7 - 1299q8 - 459q9 + 1785q10 + 531q11 - 356q12 - 1071q13 - 689q14 + 1179q15 + 529q16 + 73q17 - 681q18 - 725q19 + 584q20 + 351q21 + 292q22 - 286q23 - 548q24 + 197q25 + 127q26 + 273q27 - 50q28 - 302q29 + 50q30 - 2q31 + 153q32 + 19q33 - 129q34 + 20q35 - 25q36 + 61q37 + 15q38 - 47q39 + 12q40 - 13q41 + 18q42 + 6q43 - 14q44 + 4q45 - 3q46 + 4q47 + q48 - 3q49 + q50

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, 33]]
Out[2]=	PD[X[6, 2, 7, 1], X[14, 6, 15, 5], X[20, 15, 1, 16], X[16, 7, 17, 8], > X[8, 19, 9, 20], X[18, 9, 19, 10], X[10, 17, 11, 18], X[2, 14, 3, 13], > X[12, 4, 13, 3], X[4, 12, 5, 11]]
In[3]:=	GaussCode[Knot[10, 33]]
Out[3]=	GaussCode[1, -8, 9, -10, 2, -1, 4, -5, 6, -7, 10, -9, 8, -2, 3, -4, 7, -6, 5, > -3]
In[4]:=	DTCode[Knot[10, 33]]
Out[4]=	DTCode[6, 12, 14, 16, 18, 4, 2, 20, 10, 8]
In[5]:=	br = BR[Knot[10, 33]]
Out[5]=	BR[5, {-1, -1, -2, 1, -2, 3, -2, 3, 3, 4, -3, 4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 33]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 33]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 33]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{FullyAmphicheiral, 1, 2, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 33]][t]
Out[10]=	4 16 2 25 + -- - -- - 16 t + 4 t 2 t t
In[11]:=	Conway[Knot[10, 33]][z]
Out[11]=	4 1 + 4 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 33], Knot[11, Alternating, 333]}
In[13]:=	{KnotDet[Knot[10, 33]], KnotSignature[Knot[10, 33]]}
Out[13]=	{65, 0}
In[14]:=	Jones[Knot[10, 33]][q]
Out[14]=	-5 3 5 8 10 2 3 4 5 11 - q + -- - -- + -- - -- - 10 q + 8 q - 5 q + 3 q - q 4 3 2 q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 33]}
In[16]:=	A2Invariant[Knot[10, 33]][q]
Out[16]=	-16 -14 -12 2 2 -4 2 2 4 8 10 -1 - q + q + q - --- + -- - q + -- + 2 q - q + 2 q - 2 q + 10 8 2 q q q 12 14 16 > q + q - q
In[17]:=	HOMFLYPT[Knot[10, 33]][a, z]
Out[17]=	2 4 2 z 4 2 4 z 2 4 1 + 2 z - -- - a z + 2 z + -- + a z 4 2 a a
In[18]:=	Kauffman[Knot[10, 33]][a, z]
Out[18]=	2 3 3 3 2 z 6 z 3 2 3 z 4 2 2 z 6 z 18 z 1 - --- - --- - 6 a z - 2 a z - 6 z + ---- + 3 a z - ---- + ---- + ----- + 3 a 4 5 3 a a a a a 4 4 5 3 3 3 5 3 4 7 z z 2 4 4 4 z > 18 a z + 6 a z - 2 a z + 16 z - ---- + -- + a z - 7 a z + -- - 4 2 5 a a a 5 5 6 6 9 z 16 z 5 3 5 5 5 6 3 z 4 z 2 6 > ---- - ----- - 16 a z - 9 a z + a z - 14 z + ---- - ---- - 4 a z + 3 a 4 2 a a a 7 7 8 9 4 6 4 z 5 z 7 3 7 8 3 z 2 8 z 9 > 3 a z + ---- + ---- + 5 a z + 4 a z + 6 z + ---- + 3 a z + -- + a z 3 a 2 a a a
In[19]:=	{Vassiliev[2][Knot[10, 33]], Vassiliev[3][Knot[10, 33]]}
Out[19]=	{0, 0}
In[20]:=	Kh[Knot[10, 33]][q, t]
Out[20]=	6 1 2 1 3 2 5 3 5 5 - + 6 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 9 4 > 5 q t + 5 q t + 3 q t + 5 q t + 2 q t + 3 q t + q t + 2 q t + 11 5 > q t
In[21]:=	ColouredJones[Knot[10, 33], 2][q]
Out[21]=	-15 3 -13 8 14 27 31 9 58 48 28 89 55 103 + q - --- + q + --- - --- + -- - -- - -- + -- - -- - -- + -- - -- - 14 12 11 9 8 7 6 5 4 3 2 q q q q q q q q q q q 47 2 3 4 5 6 7 8 9 > -- - 47 q - 55 q + 89 q - 28 q - 48 q + 58 q - 9 q - 31 q + 27 q - q 11 12 13 14 15 > 14 q + 8 q + q - 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1033

1032 1034