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1015 1017

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KnotPlot

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 14 12 16 18 20 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:

- 4t-2 + 12t-1 - 15 + 12t - 4t2

Conway Polynomial:

1 - 4z2 - 4z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{47, 2}

Jones Polynomial:

q-3 - 2q-2 + 4q-1 - 5 + 7q - 8q2 + 7q3 - 6q4 + 4q5 - 2q6 + q7

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

{...}

A2 (sl(3)) Invariant:

q-10 + 2q-4 + 1 + q2 - 2q4 - 2q8 - q14 + 2q16 + q22

HOMFLY-PT Polynomial:

a-6 + a-6z2 - a-4z2 - a-4z4 - 2a-2 - 4a-2z2 - 2a-2z4 + 1 - z2 - z4 + a2 + a2z2

Kauffman Polynomial:

- 2a-8z2 + a-8z4 - 3a-7z3 + 2a-7z5 - a-6 + 5a-6z2 - 6a-6z4 + 3a-6z6 - 4a-5z + 10a-5z3 - 7a-5z5 + 3a-5z7 + 2a-4z2 + 2a-4z4 - 3a-4z6 + 2a-4z8 - 4a-3z + 8a-3z3 - 4a-3z5 + a-3z9 + 2a-2 - 11a-2z2 + 17a-2z4 - 13a-2z6 + 4a-2z8 - 2a-1z5 - a-1z7 + a-1z9 + 1 - 2z2 + 4z4 - 6z6 + 2z8 + 5az3 - 7az5 + 2az7 - a2 + 4a2z2 - 4a2z4 + a2z6

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

{-4, -4}

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 1016. 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 = 15                     1 j = 13                   1   j = 11                 3 1   j = 9               3 1     j = 7             4 3       j = 5           4 3         j = 3         3 4           j = 1       3 5             j = -1     1 2               j = -3   1 3                 j = -5   1                   j = -7 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-10 - 2q-9 + 6q-7 - 7q-6 - 4q-5 + 17q-4 - 11q-3 - 14q-2 + 29q-1 - 10 - 27q + 38q2 - 5q3 - 38q4 + 41q5 + 2q6 - 41q7 + 35q8 + 5q9 - 32q10 + 23q11 + 4q12 - 18q13 + 11q14 + 2q15 - 8q16 + 4q17 + q18 - 2q19 + q20
3	q-21 - 2q-20 + 2q-18 + 3q-17 - 6q-16 - 5q-15 + 8q-14 + 13q-13 - 13q-12 - 19q-11 + 9q-10 + 35q-9 - 9q-8 - 43q-7 - 4q-6 + 56q-5 + 13q-4 - 57q-3 - 33q-2 + 62q-1 + 44 - 52q - 62q2 + 48q3 + 71q4 - 34q5 - 84q6 + 24q7 + 92q8 - 13q9 - 95q10 + 97q12 + 7q13 - 89q14 - 16q15 + 79q16 + 16q17 - 60q18 - 19q19 + 47q20 + 12q21 - 30q22 - 8q23 + 19q24 + 5q25 - 14q26 + q27 + 8q28 + q29 - 9q30 + 2q31 + 4q32 + q33 - 5q34 + q35 + q36 + q37 - 2q38 + q39

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, 16]]
Out[2]=	PD[X[6, 2, 7, 1], X[16, 8, 17, 7], X[12, 5, 13, 6], X[14, 3, 15, 4], > X[4, 13, 5, 14], X[2, 15, 3, 16], X[20, 12, 1, 11], X[8, 20, 9, 19], > X[18, 10, 19, 9], X[10, 18, 11, 17]]
In[3]:=	GaussCode[Knot[10, 16]]
Out[3]=	GaussCode[1, -6, 4, -5, 3, -1, 2, -8, 9, -10, 7, -3, 5, -4, 6, -2, 10, -9, 8, > -7]
In[4]:=	DTCode[Knot[10, 16]]
Out[4]=	DTCode[6, 14, 12, 16, 18, 20, 4, 2, 10, 8]
In[5]:=	br = BR[Knot[10, 16]]
Out[5]=	BR[5, {1, 1, 2, -1, 2, 2, -3, 2, -3, -4, 3, -4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 16]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 16]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 16]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 16]][t]
Out[10]=	4 12 2 -15 - -- + -- + 12 t - 4 t 2 t t
In[11]:=	Conway[Knot[10, 16]][z]
Out[11]=	2 4 1 - 4 z - 4 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 16]}
In[13]:=	{KnotDet[Knot[10, 16]], KnotSignature[Knot[10, 16]]}
Out[13]=	{47, 2}
In[14]:=	Jones[Knot[10, 16]][q]
Out[14]=	-3 2 4 2 3 4 5 6 7 -5 + q - -- + - + 7 q - 8 q + 7 q - 6 q + 4 q - 2 q + q 2 q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 16]}
In[16]:=	A2Invariant[Knot[10, 16]][q]
Out[16]=	-10 2 2 4 8 14 16 22 1 + q + -- + q - 2 q - 2 q - q + 2 q + q 4 q
In[17]:=	HOMFLYPT[Knot[10, 16]][a, z]
Out[17]=	2 2 2 4 4 -6 2 2 2 z z 4 z 2 2 4 z 2 z 1 + a - -- + a - z + -- - -- - ---- + a z - z - -- - ---- 2 6 4 2 4 2 a a a a a a
In[18]:=	Kauffman[Knot[10, 16]][a, z]
Out[18]=	2 2 2 2 -6 2 2 4 z 4 z 2 2 z 5 z 2 z 11 z 2 2 1 - a + -- - a - --- - --- - 2 z - ---- + ---- + ---- - ----- + 4 a z - 2 5 3 8 6 4 2 a a a a a a a 3 3 3 4 4 4 4 3 z 10 z 8 z 3 4 z 6 z 2 z 17 z 2 4 > ---- + ----- + ---- + 5 a z + 4 z + -- - ---- + ---- + ----- - 4 a z + 7 5 3 8 6 4 2 a a a a a a a 5 5 5 5 6 6 6 2 z 7 z 4 z 2 z 5 6 3 z 3 z 13 z 2 6 > ---- - ---- - ---- - ---- - 7 a z - 6 z + ---- - ---- - ----- + a z + 7 5 3 a 6 4 2 a a a a a a 7 7 8 8 9 9 3 z z 7 8 2 z 4 z z z > ---- - -- + 2 a z + 2 z + ---- + ---- + -- + -- 5 a 4 2 3 a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 16]], Vassiliev[3][Knot[10, 16]]}
Out[19]=	{-4, -4}
In[20]:=	Kh[Knot[10, 16]][q, t]
Out[20]=	3 1 1 1 3 1 2 3 q 3 5 q + 3 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 4 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 4 q t + 3 q t + 4 q t + 3 q t + 3 q t + q t + 3 q t + 11 5 13 5 15 6 > q t + q t + q t
In[21]:=	ColouredJones[Knot[10, 16], 2][q]
Out[21]=	-10 2 6 7 4 17 11 14 29 2 3 -10 + q - -- + -- - -- - -- + -- - -- - -- + -- - 27 q + 38 q - 5 q - 9 7 6 5 4 3 2 q q q q q q q q 4 5 6 7 8 9 10 11 12 > 38 q + 41 q + 2 q - 41 q + 35 q + 5 q - 32 q + 23 q + 4 q - 13 14 15 16 17 18 19 20 > 18 q + 11 q + 2 q - 8 q + 4 q + q - 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1016

1015 1017