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

1064 1066

KnotPlot

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KnotPlot

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 16 2 18 20 8 6 12

Minimum Braid Representative: Length is 11, width is 4 Braid index is 4

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index Reversible 2 3 3 / NotAvailable 1

Alexander Polynomial:

2t-3 - 7t-2 + 14t-1 - 17 + 14t - 7t2 + 2t3

Conway Polynomial:

1 + 4z2 + 5z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{1077, K11n71, K11n75, ...}

Determinant and Signature:

{63, 2}

Jones Polynomial:

- q-2 + 3q-1 - 5 + 8q - 10q2 + 11q3 - 9q4 + 8q5 - 5q6 + 2q7 - q8

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

{...}

A2 (sl(3)) Invariant:

- q-6 + q-4 + 2q2 - 3q4 + q6 + 2q10 + 4q12 + 2q16 - 2q18 - 2q20 - q24

HOMFLY-PT Polynomial:

- 3a-6 - 3a-6z2 - a-6z4 + 5a-4 + 7a-4z2 + 4a-4z4 + a-4z6 - a-2 + 2a-2z2 + 3a-2z4 + a-2z6 - 2z2 - z4

Kauffman Polynomial:

2a-9z - 3a-9z3 + a-9z5 + a-8z2 - 4a-8z4 + 2a-8z6 - 2a-7z + 4a-7z3 - 6a-7z5 + 3a-7z7 + 3a-6 - 12a-6z2 + 12a-6z4 - 7a-6z6 + 3a-6z8 - 8a-5z + 19a-5z3 - 14a-5z5 + 4a-5z7 + a-5z9 + 5a-4 - 17a-4z2 + 24a-4z4 - 16a-4z6 + 6a-4z8 - 6a-3z + 20a-3z3 - 17a-3z5 + 5a-3z7 + a-3z9 + a-2 - a-2z2 + a-2z4 - 4a-2z6 + 3a-2z8 - 2a-1z + 6a-1z3 - 9a-1z5 + 4a-1z7 + 3z2 - 7z4 + 3z6 - 2az3 + az5

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

{4, 7}

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 1065. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                     1 j = 15                   1   j = 13                 4 1   j = 11               4 1     j = 9             5 4       j = 7           6 4         j = 5         4 5           j = 3       4 6             j = 1     2 5               j = -1   1 3                 j = -3   2                   j = -5 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-7 - 3q-6 + q-5 + 8q-4 - 14q-3 + 27q-1 - 31 - 10q + 59q2 - 46q3 - 31q4 + 90q5 - 50q6 - 52q7 + 100q8 - 40q9 - 58q10 + 84q11 - 20q12 - 49q13 + 52q14 - 4q15 - 30q16 + 21q17 + 2q18 - 11q19 + 5q20 + q21 - 2q22 + q23
3	- q-15 + 3q-14 - q-13 - 4q-12 - q-11 + 11q-10 + 2q-9 - 21q-8 - 6q-7 + 35q-6 + 16q-5 - 55q-4 - 34q-3 + 75q-2 + 69q-1 - 98 - 112q + 105q2 + 178q3 - 110q4 - 237q5 + 88q6 + 308q7 - 64q8 - 360q9 + 24q10 + 405q11 + 11q12 - 418q13 - 59q14 + 429q15 + 84q16 - 399q17 - 127q18 + 371q19 + 140q20 - 307q21 - 166q22 + 252q23 + 164q24 - 179q25 - 159q26 + 119q27 + 136q28 - 66q29 - 105q30 + 25q31 + 78q32 - 11q33 - 42q34 - 5q35 + 27q36 + q37 - 10q38 - 3q39 + 7q40 - q41 - q42 - q43 + 2q44 - q45
4	q-26 - 3q-25 + q-24 + 4q-23 - 3q-22 + 4q-21 - 14q-20 + 6q-19 + 18q-18 - 13q-17 + 12q-16 - 47q-15 + 14q-14 + 62q-13 - 22q-12 + 20q-11 - 134q-10 + 10q-9 + 158q-8 + 17q-7 + 61q-6 - 319q-5 - 95q-4 + 264q-3 + 184q-2 + 266q-1 - 559 - 410q + 204q2 + 435q3 + 776q4 - 649q5 - 881q6 - 183q7 + 554q8 + 1504q9 - 424q10 - 1265q11 - 802q12 + 398q13 + 2164q14 + 15q15 - 1390q16 - 1372q17 + 57q18 + 2525q19 + 454q20 - 1269q21 - 1718q22 - 309q23 + 2557q24 + 779q25 - 987q26 - 1812q27 - 639q28 + 2277q29 + 979q30 - 555q31 - 1660q32 - 929q33 + 1716q34 + 1020q35 - 34q36 - 1251q37 - 1082q38 + 981q39 + 829q40 + 384q41 - 673q42 - 963q43 + 328q44 + 446q45 + 499q46 - 164q47 - 608q48 - 11q49 + 94q50 + 340q51 + 66q52 - 251q53 - 51q54 - 56q55 + 133q56 + 71q57 - 66q58 - 4q59 - 49q60 + 32q61 + 24q62 - 18q63 + 11q64 - 16q65 + 6q66 + 5q67 - 7q68 + 5q69 - 3q70 + q71 + q72 - 2q73 + q74

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, 65]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[11, 19, 12, 18], X[5, 15, 6, 14], > X[17, 7, 18, 6], X[7, 17, 8, 16], X[15, 9, 16, 8], X[13, 1, 14, 20], > X[19, 13, 20, 12], X[9, 2, 10, 3]]
In[3]:=	GaussCode[Knot[10, 65]]
Out[3]=	GaussCode[-1, 10, -2, 1, -4, 5, -6, 7, -10, 2, -3, 9, -8, 4, -7, 6, -5, 3, -9, > 8]
In[4]:=	DTCode[Knot[10, 65]]
Out[4]=	DTCode[4, 10, 14, 16, 2, 18, 20, 8, 6, 12]
In[5]:=	br = BR[Knot[10, 65]]
Out[5]=	BR[4, {1, 1, 2, -1, 2, -3, 2, 2, 2, -3, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 65]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 65]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 65]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 65]][t]
Out[10]=	2 7 14 2 3 -17 + -- - -- + -- + 14 t - 7 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 65]][z]
Out[11]=	2 4 6 1 + 4 z + 5 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 65], Knot[10, 77], Knot[11, NonAlternating, 71], > Knot[11, NonAlternating, 75]}
In[13]:=	{KnotDet[Knot[10, 65]], KnotSignature[Knot[10, 65]]}
Out[13]=	{63, 2}
In[14]:=	Jones[Knot[10, 65]][q]
Out[14]=	-2 3 2 3 4 5 6 7 8 -5 - q + - + 8 q - 10 q + 11 q - 9 q + 8 q - 5 q + 2 q - q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 65]}
In[16]:=	A2Invariant[Knot[10, 65]][q]
Out[16]=	-6 -4 2 4 6 10 12 16 18 20 24 -q + q + 2 q - 3 q + q + 2 q + 4 q + 2 q - 2 q - 2 q - q
In[17]:=	HOMFLYPT[Knot[10, 65]][a, z]
Out[17]=	2 2 2 4 4 4 6 6 -3 5 -2 2 3 z 7 z 2 z 4 z 4 z 3 z z z -- + -- - a - 2 z - ---- + ---- + ---- - z - -- + ---- + ---- + -- + -- 6 4 6 4 2 6 4 2 4 2 a a a a a a a a a a
In[18]:=	Kauffman[Knot[10, 65]][a, z]
Out[18]=	2 2 2 2 3 5 -2 2 z 2 z 8 z 6 z 2 z 2 z 12 z 17 z z -- + -- + a + --- - --- - --- - --- - --- + 3 z + -- - ----- - ----- - -- - 6 4 9 7 5 3 a 8 6 4 2 a a a a a a a a a a 3 3 3 3 3 4 4 4 3 z 4 z 19 z 20 z 6 z 3 4 4 z 12 z 24 z > ---- + ---- + ----- + ----- + ---- - 2 a z - 7 z - ---- + ----- + ----- + 9 7 5 3 a 8 6 4 a a a a a a a 4 5 5 5 5 5 6 6 6 z z 6 z 14 z 17 z 9 z 5 6 2 z 7 z 16 z > -- + -- - ---- - ----- - ----- - ---- + a z + 3 z + ---- - ---- - ----- - 2 9 7 5 3 a 8 6 4 a a a a a a a a 6 7 7 7 7 8 8 8 9 9 4 z 3 z 4 z 5 z 4 z 3 z 6 z 3 z z z > ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + -- + -- 2 7 5 3 a 6 4 2 5 3 a a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 65]], Vassiliev[3][Knot[10, 65]]}
Out[19]=	{4, 7}
In[20]:=	Kh[Knot[10, 65]][q, t]
Out[20]=	3 1 2 1 3 2 q 3 5 5 2 5 q + 4 q + ----- + ----- + ---- + --- + --- + 6 q t + 4 q t + 5 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 13 5 > 6 q t + 4 q t + 5 q t + 4 q t + 4 q t + q t + 4 q t + 13 6 15 6 17 7 > q t + q t + q t
In[21]:=	ColouredJones[Knot[10, 65], 2][q]
Out[21]=	-7 3 -5 8 14 27 2 3 4 5 -31 + q - -- + q + -- - -- + -- - 10 q + 59 q - 46 q - 31 q + 90 q - 6 4 3 q q q q 6 7 8 9 10 11 12 13 > 50 q - 52 q + 100 q - 40 q - 58 q + 84 q - 20 q - 49 q + 14 15 16 17 18 19 20 21 22 > 52 q - 4 q - 30 q + 21 q + 2 q - 11 q + 5 q + q - 2 q + 23 > q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1065

1064 1066