© | Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table:

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KnotPlot

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KnotPlot

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 12 14 20 18 16 4 2 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 Reversible 2 2 2 / NotAvailable 1

Alexander Polynomial:

- 3t-2 + 7t-1 - 7 + 7t - 3t2

Conway Polynomial:

1 - 5z2 - 3z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{27, -2}

Jones Polynomial:

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

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

{...}

A2 (sl(3)) Invariant:

q-16 + q-10 + q-6 - q2 - q4 - q6 - q8 + q10 + q12 + q14 + q16

HOMFLY-PT Polynomial:

2a-4 + a-4z2 - 2a-2 - 3a-2z2 - a-2z4 - 2z2 - z4 - 2a2z2 - a2z4 + a4 + a4z2

Kauffman Polynomial:

2a-4 - 13a-4z2 + 16a-4z4 - 7a-4z6 + a-4z8 + 2a-3z - 7a-3z3 + 11a-3z5 - 6a-3z7 + a-3z9 + 2a-2 - 16a-2z2 + 29a-2z4 - 17a-2z6 + 3a-2z8 - a-1z + 7a-1z3 - 2a-1z5 - 3a-1z7 + a-1z9 + z2 + 4z4 - 7z6 + 2z8 - 3az + 8az3 - 10az5 + 3az7 - 6a2z4 + 3a2z6 - 4a3z3 + 3a3z5 + a4 - 3a4z2 + 3a4z4 + 2a5z3 + a6z2

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

{-5, -1}

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

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

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-14 - 2q-13 + q-12 + 3q-11 - 5q-10 + 2q-9 + 3q-8 - 6q-7 + 3q-6 + 3q-5 - 6q-4 + 3q-3 + 4q-2 - 8q-1 + 4 + 5q - 9q2 + 4q3 + 6q4 - 8q5 + 2q6 + 6q7 - 7q8 + 5q10 - 4q11 - q12 + 3q13 - q14 - q15 + q16
3	q-27 - 2q-26 + q-25 + q-24 + q-23 - 4q-22 + 4q-20 + q-19 - 6q-18 - q-17 + 7q-16 + 3q-15 - 10q-14 - 3q-13 + 9q-12 + 8q-11 - 13q-10 - 7q-9 + 11q-8 + 11q-7 - 12q-6 - 10q-5 + 10q-4 + 10q-3 - 9q-2 - 8q-1 + 9 + 5q - 7q2 - 5q3 + 8q4 + 2q5 - 6q6 - 3q7 + 7q8 + q9 - 6q10 - q11 + 6q12 + q13 - 7q14 + 7q16 + 2q17 - 8q18 - q19 + 6q20 + 4q21 - 7q22 - 3q23 + 3q24 + 5q25 - 3q26 - 3q27 + 3q29 - q31 - q32 + q33
4	q-44 - 2q-43 + q-42 + q-41 - q-40 + 2q-39 - 6q-38 + 4q-37 + 2q-36 - 2q-35 + 3q-34 - 11q-33 + 10q-32 + 3q-31 - 5q-30 - 14q-28 + 20q-27 + 5q-26 - 9q-25 - 7q-24 - 16q-23 + 31q-22 + 9q-21 - 14q-20 - 15q-19 - 19q-18 + 41q-17 + 14q-16 - 16q-15 - 22q-14 - 26q-13 + 47q-12 + 19q-11 - 12q-10 - 23q-9 - 33q-8 + 43q-7 + 20q-6 - 6q-5 - 17q-4 - 36q-3 + 36q-2 + 17q-1 - 4 - 9q - 34q2 + 30q3 + 14q4 - 3q5 - 4q6 - 33q7 + 24q8 + 13q9 - q10 + 2q11 - 32q12 + 16q13 + 10q14 + q15 + 9q16 - 30q17 + 10q18 + 6q19 + 13q21 - 24q22 + 7q23 + 4q24 - 3q25 + 12q26 - 19q27 + 7q28 + 5q29 - 4q30 + 8q31 - 17q32 + 6q33 + 7q34 + 6q36 - 16q37 + q38 + 5q39 + 3q40 + 8q41 - 11q42 - 3q43 + 2q45 + 8q46 - 4q47 - 2q48 - 2q49 - q50 + 4q51 - q54 - q55 + q56

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

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 104

103 105