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1067 1069

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PD Presentation:

X4251 X12,4,13,3 X20,13,1,14 X16,5,17,6 X8,19,9,20 X18,9,19,10 X10,17,11,18 X14,7,15,8 X6,15,7,16 X2,12,3,11

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 16 14 18 2 20 6 10 8

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 2 3 / NotAvailable 1

Alexander Polynomial:

4t-2 - 14t-1 + 21 - 14t + 4t2

Conway Polynomial:

1 + 2z2 + 4z4

Other knots with the same Alexander/Conway Polynomial:

{1031, ...}

Determinant and Signature:

{57, 0}

Jones Polynomial:

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

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

{...}

A2 (sl(3)) Invariant:

- q-22 - 2q-16 + 2q-14 + q-12 + 2q-8 - q-6 + q-4 + 2q2 - 2q4 + q6 + q8 - q10

HOMFLY-PT Polynomial:

- a-2z2 + z4 + a2 + 3a2z2 + 2a2z4 + a4 + a4z2 + a4z4 - a6 - a6z2

Kauffman Polynomial:

a-3z3 - a-2z2 + 3a-2z4 - 3a-1z3 + 5a-1z5 + 4z2 - 10z4 + 7z6 - 2az + 8az3 - 14az5 + 7az7 - a2 + 7a2z2 - 9a2z4 - 4a2z6 + 4a2z8 - 6a3z + 27a3z3 - 30a3z5 + 7a3z7 + a3z9 + a4 - 5a4z2 + 17a4z4 - 20a4z6 + 6a4z8 - 8a5z + 23a5z3 - 16a5z5 + a5z7 + a5z9 + a6 - 7a6z2 + 13a6z4 - 9a6z6 + 2a6z8 - 4a7z + 8a7z3 - 5a7z5 + a7z7

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

{2, -3}

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 1068. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-21 - 2q-20 - q-19 + 7q-18 - 6q-17 - 9q-16 + 21q-15 - 6q-14 - 27q-13 + 36q-12 + 3q-11 - 49q-10 + 41q-9 + 19q-8 - 64q-7 + 36q-6 + 34q-5 - 64q-4 + 24q-3 + 38q-2 - 51q-1 + 12 + 28q - 30q2 + 6q3 + 12q4 - 12q5 + 4q6 + 2q7 - 3q8 + q9
3	- q-42 + 2q-41 + q-40 - 2q-39 - 6q-38 + 5q-37 + 11q-36 - 2q-35 - 24q-34 - 2q-33 + 34q-32 + 19q-31 - 47q-30 - 39q-29 + 47q-28 + 69q-27 - 40q-26 - 96q-25 + 22q-24 + 115q-23 + 11q-22 - 132q-21 - 41q-20 + 130q-19 + 80q-18 - 127q-17 - 108q-16 + 104q-15 + 144q-14 - 87q-13 - 163q-12 + 54q-11 + 185q-10 - 25q-9 - 191q-8 - 10q-7 + 192q-6 + 34q-5 - 174q-4 - 57q-3 + 153q-2 + 60q-1 - 114 - 62q + 88q2 + 43q3 - 52q4 - 32q5 + 34q6 + 13q7 - 16q8 - 6q9 + 10q10 - 2q11 - 3q12 + 3q13 - q15 - 2q16 + 3q17 - q18

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, 68]]
Out[2]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[20, 13, 1, 14], X[16, 5, 17, 6], > X[8, 19, 9, 20], X[18, 9, 19, 10], X[10, 17, 11, 18], X[14, 7, 15, 8], > X[6, 15, 7, 16], X[2, 12, 3, 11]]
In[3]:=	GaussCode[Knot[10, 68]]
Out[3]=	GaussCode[1, -10, 2, -1, 4, -9, 8, -5, 6, -7, 10, -2, 3, -8, 9, -4, 7, -6, 5, > -3]
In[4]:=	DTCode[Knot[10, 68]]
Out[4]=	DTCode[4, 12, 16, 14, 18, 2, 20, 6, 10, 8]
In[5]:=	br = BR[Knot[10, 68]]
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[10, 68]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 68]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 68]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 68]][t]
Out[10]=	4 14 2 21 + -- - -- - 14 t + 4 t 2 t t
In[11]:=	Conway[Knot[10, 68]][z]
Out[11]=	2 4 1 + 2 z + 4 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 31], Knot[10, 68]}
In[13]:=	{KnotDet[Knot[10, 68]], KnotSignature[Knot[10, 68]]}
Out[13]=	{57, 0}
In[14]:=	Jones[Knot[10, 68]][q]
Out[14]=	-7 2 4 7 8 9 9 2 3 8 - q + -- - -- + -- - -- + -- - - - 5 q + 3 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, 68]}
In[16]:=	A2Invariant[Knot[10, 68]][q]
Out[16]=	-22 2 2 -12 2 -6 -4 2 4 6 8 10 -q - --- + --- + q + -- - q + q + 2 q - 2 q + q + q - q 16 14 8 q q q
In[17]:=	HOMFLYPT[Knot[10, 68]][a, z]
Out[17]=	2 2 4 6 z 2 2 4 2 6 2 4 2 4 4 4 a + a - a - -- + 3 a z + a z - a z + z + 2 a z + a z 2 a
In[18]:=	Kauffman[Knot[10, 68]][a, z]
Out[18]=	2 2 4 6 3 5 7 2 z 2 2 -a + a + a - 2 a z - 6 a z - 8 a z - 4 a z + 4 z - -- + 7 a z - 2 a 3 3 4 2 6 2 z 3 z 3 3 3 5 3 7 3 > 5 a z - 7 a z + -- - ---- + 8 a z + 27 a z + 23 a z + 8 a z - 3 a a 4 5 4 3 z 2 4 4 4 6 4 5 z 5 3 5 > 10 z + ---- - 9 a z + 17 a z + 13 a z + ---- - 14 a z - 30 a z - 2 a a 5 5 7 5 6 2 6 4 6 6 6 7 > 16 a z - 5 a z + 7 z - 4 a z - 20 a z - 9 a z + 7 a z + 3 7 5 7 7 7 2 8 4 8 6 8 3 9 5 9 > 7 a z + a z + a z + 4 a z + 6 a z + 2 a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[10, 68]], Vassiliev[3][Knot[10, 68]]}
Out[19]=	{2, -3}
In[20]:=	Kh[Knot[10, 68]][q, t]
Out[20]=	4 1 1 1 3 1 4 3 4 - + 5 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 7 3 q t q t q t q t q t q t q t q t 4 5 4 4 5 3 3 2 5 2 > ----- + ----- + ----- + ---- + --- + 2 q t + 3 q t + q t + 2 q t + 5 3 5 2 3 2 3 q t q t q t q t q t 7 3 > q t
In[21]:=	ColouredJones[Knot[10, 68], 2][q]
Out[21]=	-21 2 -19 7 6 9 21 6 27 36 3 49 12 + q - --- - q + --- - --- - --- + --- - --- - --- + --- + --- - --- + 20 18 17 16 15 14 13 12 11 10 q q q q q q q q q q 41 19 64 36 34 64 24 38 51 2 3 4 > -- + -- - -- + -- + -- - -- + -- + -- - -- + 28 q - 30 q + 6 q + 12 q - 9 8 7 6 5 4 3 2 q q q q q q q q q 5 6 7 8 9 > 12 q + 4 q + 2 q - 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1068

1067 1069