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

1017 1019

KnotPlot

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KnotPlot

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 14 18 16 2 10 20 8 6

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 1 2 2 / NotAvailable 1

Alexander Polynomial:

- 4t-2 + 14t-1 - 19 + 14t - 4t2

Conway Polynomial:

1 - 2z2 - 4z4

Other knots with the same Alexander/Conway Polynomial:

{1024, ...}

Determinant and Signature:

{55, -2}

Jones Polynomial:

q-7 - 3q-6 + 5q-5 - 7q-4 + 9q-3 - 9q-2 + 8q-1 - 6 + 4q - 2q2 + q3

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

{...}

A2 (sl(3)) Invariant:

q-22 - q-20 - q-18 + 2q-16 - q-14 + q-12 + q-10 - q-8 + q-6 - 2q-4 + q-2 - q2 + 2q4 + q10

HOMFLY-PT Polynomial:

a-2 + a-2z2 - z2 - z4 - a2 - 3a2z2 - 2a2z4 + a4 - a4z4 + a6z2

Kauffman Polynomial:

- a-2 + 4a-2z2 - 4a-2z4 + a-2z6 - 2a-1z + 6a-1z3 - 7a-1z5 + 2a-1z7 + z2 + z4 - 5z6 + 2z8 - 4az + 11az3 - 10az5 + az7 + az9 + a2 - 8a2z2 + 17a2z4 - 15a2z6 + 5a2z8 - 4a3z + 14a3z3 - 12a3z5 + 3a3z7 + a3z9 + a4 - 3a4z2 + 6a4z4 - 5a4z6 + 3a4z8 - 2a5z + 5a5z3 - 6a5z5 + 4a5z7 + a6z2 - 5a6z4 + 4a6z6 - 4a7z3 + 3a7z5 - a8z2 + a8z4

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

{-2, 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 1018. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-20 - 3q-19 + q-18 + 7q-17 - 12q-16 + 3q-15 + 19q-14 - 29q-13 + 4q-12 + 38q-11 - 48q-10 + q-9 + 57q-8 - 57q-7 - 6q-6 + 63q-5 - 50q-4 - 14q-3 + 55q-2 - 32q-1 - 18 + 38q - 14q2 - 16q3 + 19q4 - 3q5 - 8q6 + 6q7 - 2q9 + q10
3	q-39 - 3q-38 + q-37 + 3q-36 + 2q-35 - 7q-34 - 3q-33 + 11q-32 + 2q-31 - 15q-30 - 2q-29 + 25q-28 - 2q-27 - 38q-26 + 4q-25 + 61q-24 - 6q-23 - 89q-22 + 3q-21 + 119q-20 + 8q-19 - 150q-18 - 20q-17 + 168q-16 + 43q-15 - 186q-14 - 56q-13 + 182q-12 + 78q-11 - 176q-10 - 92q-9 + 158q-8 + 107q-7 - 136q-6 - 116q-5 + 106q-4 + 124q-3 - 77q-2 - 122q-1 + 44 + 117q - 17q2 - 100q3 - 7q4 + 81q5 + 19q6 - 56q7 - 27q8 + 39q9 + 21q10 - 19q11 - 19q12 + 12q13 + 10q14 - 4q15 - 7q16 + 3q17 + 2q18 - 2q20 + q21
4	q-64 - 3q-63 + q-62 + 3q-61 - 2q-60 + 7q-59 - 13q-58 + q-57 + 5q-56 - 8q-55 + 32q-54 - 28q-53 + 2q-52 - 36q-50 + 70q-49 - 33q-48 + 32q-47 - 5q-46 - 112q-45 + 83q-44 - 39q-43 + 130q-42 + 47q-41 - 225q-40 + 8q-39 - 109q-38 + 291q-37 + 217q-36 - 295q-35 - 143q-34 - 305q-33 + 422q-32 + 474q-31 - 253q-30 - 266q-29 - 574q-28 + 439q-27 + 683q-26 - 125q-25 - 273q-24 - 790q-23 + 355q-22 + 756q-21 + 2q-20 - 184q-19 - 874q-18 + 230q-17 + 696q-16 + 95q-15 - 43q-14 - 846q-13 + 84q-12 + 548q-11 + 168q-10 + 127q-9 - 732q-8 - 75q-7 + 328q-6 + 205q-5 + 307q-4 - 534q-3 - 195q-2 + 74q-1 + 158 + 425q - 275q2 - 202q3 - 127q4 + 29q5 + 407q6 - 52q7 - 101q8 - 187q9 - 91q10 + 267q11 + 48q12 + 10q13 - 127q14 - 120q15 + 118q16 + 40q17 + 48q18 - 46q19 - 77q20 + 37q21 + 9q22 + 31q23 - 8q24 - 31q25 + 12q26 - 2q27 + 10q28 - 9q30 + 4q31 - q32 + 2q33 - 2q35 + q36

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, 18]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 12, 4, 13], X[5, 14, 6, 15], X[15, 20, 16, 1], > X[9, 17, 10, 16], X[7, 19, 8, 18], X[17, 9, 18, 8], X[19, 7, 20, 6], > X[13, 10, 14, 11], X[11, 2, 12, 3]]
In[3]:=	GaussCode[Knot[10, 18]]
Out[3]=	GaussCode[-1, 10, -2, 1, -3, 8, -6, 7, -5, 9, -10, 2, -9, 3, -4, 5, -7, 6, -8, > 4]
In[4]:=	DTCode[Knot[10, 18]]
Out[4]=	DTCode[4, 12, 14, 18, 16, 2, 10, 20, 8, 6]
In[5]:=	br = BR[Knot[10, 18]]
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, 18]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 18]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 18]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 2, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 18]][t]
Out[10]=	4 14 2 -19 - -- + -- + 14 t - 4 t 2 t t
In[11]:=	Conway[Knot[10, 18]][z]
Out[11]=	2 4 1 - 2 z - 4 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 18], Knot[10, 24]}
In[13]:=	{KnotDet[Knot[10, 18]], KnotSignature[Knot[10, 18]]}
Out[13]=	{55, -2}
In[14]:=	Jones[Knot[10, 18]][q]
Out[14]=	-7 3 5 7 9 9 8 2 3 -6 + q - -- + -- - -- + -- - -- + - + 4 q - 2 q + q 6 5 4 3 2 q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 18]}
In[16]:=	A2Invariant[Knot[10, 18]][q]
Out[16]=	-22 -20 -18 2 -14 -12 -10 -8 -6 2 -2 2 q - q - q + --- - q + q + q - q + q - -- + q - q + 16 4 q q 4 10 > 2 q + q
In[17]:=	HOMFLYPT[Knot[10, 18]][a, z]
Out[17]=	2 -2 2 4 2 z 2 2 6 2 4 2 4 4 4 a - a + a - z + -- - 3 a z + a z - z - 2 a z - a z 2 a
In[18]:=	Kauffman[Knot[10, 18]][a, z]
Out[18]=	2 -2 2 4 2 z 3 5 2 4 z 2 2 -a + a + a - --- - 4 a z - 4 a z - 2 a z + z + ---- - 8 a z - a 2 a 3 4 2 6 2 8 2 6 z 3 3 3 5 3 7 3 > 3 a z + a z - a z + ---- + 11 a z + 14 a z + 5 a z - 4 a z + a 4 5 4 4 z 2 4 4 4 6 4 8 4 7 z 5 > z - ---- + 17 a z + 6 a z - 5 a z + a z - ---- - 10 a z - 2 a a 6 3 5 5 5 7 5 6 z 2 6 4 6 6 6 > 12 a z - 6 a z + 3 a z - 5 z + -- - 15 a z - 5 a z + 4 a z + 2 a 7 2 z 7 3 7 5 7 8 2 8 4 8 9 3 9 > ---- + a z + 3 a z + 4 a z + 2 z + 5 a z + 3 a z + a z + a z a
In[19]:=	{Vassiliev[2][Knot[10, 18]], Vassiliev[3][Knot[10, 18]]}
Out[19]=	{-2, 1}
In[20]:=	Kh[Knot[10, 18]][q, t]
Out[20]=	4 5 1 2 1 3 2 4 3 5 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t q t 4 4 5 3 t 2 3 2 3 3 5 3 7 4 > ----- + ---- + ---- + --- + 3 q t + q t + 3 q t + q t + q t + q t 5 2 5 3 q q t q t q t
In[21]:=	ColouredJones[Knot[10, 18], 2][q]
Out[21]=	-20 3 -18 7 12 3 19 29 4 38 48 -9 -18 + q - --- + q + --- - --- + --- + --- - --- + --- + --- - --- + q + 19 17 16 15 14 13 12 11 10 q q q q q q q q q 57 57 6 63 50 14 55 32 2 3 4 > -- - -- - -- + -- - -- - -- + -- - -- + 38 q - 14 q - 16 q + 19 q - 8 7 6 5 4 3 2 q q q q q q q q 5 6 7 9 10 > 3 q - 8 q + 6 q - 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1018

1017 1019