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

1011 1013

KnotPlot

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KnotPlot

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 16 2 20 18 6 8 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 2 / NotAvailable 1

Alexander Polynomial:

2t-3 - 6t-2 + 10t-1 - 11 + 10t - 6t2 + 2t3

Conway Polynomial:

1 + 4z2 + 6z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{1054, ...}

Determinant and Signature:

{47, 2}

Jones Polynomial:

- q-2 + 2q-1 - 3 + 6q - 7q2 + 8q3 - 7q4 + 6q5 - 4q6 + 2q7 - q8

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

{...}

A2 (sl(3)) Invariant:

- q-6 + 2q2 - q4 + 2q6 + q8 + q10 + 2q12 - q14 + q16 - q18 - q20 - q24

HOMFLY-PT Polynomial:

- 2a-6 - 3a-6z2 - a-6z4 + 2a-4 + 5a-4z2 + 4a-4z4 + a-4z6 + 2a-2 + 5a-2z2 + 4a-2z4 + a-2z6 - 1 - 3z2 - z4

Kauffman Polynomial:

2a-9z - 3a-9z3 + a-9z5 + 2a-8z2 - 5a-8z4 + 2a-8z6 - a-7z3 - 3a-7z5 + 2a-7z7 + 2a-6 - 8a-6z2 + 8a-6z4 - 5a-6z6 + 2a-6z8 - 3a-5z + 4a-5z3 - a-5z7 + a-5z9 + 2a-4 - 12a-4z2 + 23a-4z4 - 14a-4z6 + 4a-4z8 - a-3z + 5a-3z3 - a-3z5 - a-3z7 + a-3z9 - 2a-2 + 2a-2z2 + 4a-2z4 - 5a-2z6 + 2a-2z8 + a-1z - 4a-1z5 + 2a-1z7 - 1 + 4z2 - 6z4 + 2z6 + az - 3az3 + az5

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

{4, 6}

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 1012. 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                 3 1   j = 11               3 1     j = 9             4 3       j = 7           4 3         j = 5         3 4           j = 3       3 4             j = 1     1 4               j = -1   1 2                 j = -3   1                   j = -5 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-7 - 2q-6 + 4q-4 - 7q-3 + q-2 + 12q-1 - 15 - 2q + 27q2 - 25q3 - 11q4 + 44q5 - 30q6 - 20q7 + 52q8 - 28q9 - 24q10 + 46q11 - 18q12 - 23q13 + 31q14 - 7q15 - 16q16 + 15q17 - q18 - 7q19 + 5q20 - 2q22 + q23
3	- q-15 + 2q-14 - q-12 - 2q-11 + 4q-10 - 5q-8 - q-7 + 10q-6 - q-5 - 16q-4 + 24q-2 + 7q-1 - 37 - 13q + 42q2 + 37q3 - 57q4 - 50q5 + 52q6 + 82q7 - 57q8 - 98q9 + 47q10 + 122q11 - 46q12 - 128q13 + 31q14 + 141q15 - 28q16 - 135q17 + 11q18 + 133q19 - 3q20 - 116q21 - 14q22 + 101q23 + 23q24 - 79q25 - 31q26 + 58q27 + 32q28 - 38q29 - 28q30 + 21q31 + 24q32 - 14q33 - 13q34 + 5q35 + 10q36 - 5q37 - 4q38 + 2q39 + 4q40 - 3q41 - q42 + 2q44 - q45
4	q-26 - 2q-25 + q-23 - q-22 + 5q-21 - 6q-20 + 2q-19 + 2q-18 - 8q-17 + 12q-16 - 11q-15 + 9q-14 + 9q-13 - 26q-12 + 14q-11 - 22q-10 + 27q-9 + 34q-8 - 47q-7 + 6q-6 - 60q-5 + 44q-4 + 86q-3 - 39q-2 + 9q-1 - 141 + 15q + 135q2 + 22q3 + 82q4 - 238q5 - 90q6 + 125q7 + 102q8 + 244q9 - 284q10 - 227q11 + 33q12 + 142q13 + 439q14 - 263q15 - 330q16 - 84q17 + 131q18 + 581q19 - 210q20 - 368q21 - 173q22 + 91q23 + 645q24 - 151q25 - 354q26 - 226q27 + 35q28 + 633q29 - 78q30 - 286q31 - 253q32 - 48q33 + 549q34 + 5q35 - 163q36 - 237q37 - 142q38 + 394q39 + 61q40 - 21q41 - 165q42 - 191q43 + 214q44 + 51q45 + 68q46 - 61q47 - 159q48 + 80q49 + q50 + 74q51 + 7q52 - 84q53 + 29q54 - 30q55 + 34q56 + 18q57 - 29q58 + 22q59 - 24q60 + 7q61 + 6q62 - 11q63 + 16q64 - 8q65 + q66 - 6q68 + 6q69 - q70 + q71 - 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, 12]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[13, 19, 14, 18], X[5, 15, 6, 14], > X[7, 17, 8, 16], X[15, 7, 16, 6], X[17, 9, 18, 8], X[11, 1, 12, 20], > X[19, 13, 20, 12], X[9, 2, 10, 3]]
In[3]:=	GaussCode[Knot[10, 12]]
Out[3]=	GaussCode[-1, 10, -2, 1, -4, 6, -5, 7, -10, 2, -8, 9, -3, 4, -6, 5, -7, 3, -9, > 8]
In[4]:=	DTCode[Knot[10, 12]]
Out[4]=	DTCode[4, 10, 14, 16, 2, 20, 18, 6, 8, 12]
In[5]:=	br = BR[Knot[10, 12]]
Out[5]=	BR[4, {1, 1, 1, 1, 1, 2, -1, -3, 2, -3, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 12]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 12]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 12]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 12]][t]
Out[10]=	2 6 10 2 3 -11 + -- - -- + -- + 10 t - 6 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 12]][z]
Out[11]=	2 4 6 1 + 4 z + 6 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 12], Knot[10, 54]}
In[13]:=	{KnotDet[Knot[10, 12]], KnotSignature[Knot[10, 12]]}
Out[13]=	{47, 2}
In[14]:=	Jones[Knot[10, 12]][q]
Out[14]=	-2 2 2 3 4 5 6 7 8 -3 - q + - + 6 q - 7 q + 8 q - 7 q + 6 q - 4 q + 2 q - q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 12]}
In[16]:=	A2Invariant[Knot[10, 12]][q]
Out[16]=	-6 2 4 6 8 10 12 14 16 18 20 24 -q + 2 q - q + 2 q + q + q + 2 q - q + q - q - q - q
In[17]:=	HOMFLYPT[Knot[10, 12]][a, z]
Out[17]=	2 2 2 4 4 4 6 6 2 2 2 2 3 z 5 z 5 z 4 z 4 z 4 z z z -1 - -- + -- + -- - 3 z - ---- + ---- + ---- - z - -- + ---- + ---- + -- + -- 6 4 2 6 4 2 6 4 2 4 2 a a a a a a a a a a a
In[18]:=	Kauffman[Knot[10, 12]][a, z]
Out[18]=	2 2 2 2 2 2 2 z 3 z z z 2 2 z 8 z 12 z -1 + -- + -- - -- + --- - --- - -- + - + a z + 4 z + ---- - ---- - ----- + 6 4 2 9 5 3 a 8 6 4 a a a a a a a a a 2 3 3 3 3 4 4 4 2 z 3 z z 4 z 5 z 3 4 5 z 8 z 23 z > ---- - ---- - -- + ---- + ---- - 3 a z - 6 z - ---- + ---- + ----- + 2 9 7 5 3 8 6 4 a a a a a a a a 4 5 5 5 5 6 6 6 6 4 z z 3 z z 4 z 5 6 2 z 5 z 14 z 5 z > ---- + -- - ---- - -- - ---- + a z + 2 z + ---- - ---- - ----- - ---- + 2 9 7 3 a 8 6 4 2 a a a a a a a a 7 7 7 7 8 8 8 9 9 2 z z z 2 z 2 z 4 z 2 z z z > ---- - -- - -- + ---- + ---- + ---- + ---- + -- + -- 7 5 3 a 6 4 2 5 3 a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 12]], Vassiliev[3][Knot[10, 12]]}
Out[19]=	{4, 6}
In[20]:=	Kh[Knot[10, 12]][q, t]
Out[20]=	3 1 1 1 2 q 3 5 5 2 4 q + 3 q + ----- + ----- + ---- + --- + - + 4 q t + 3 q t + 4 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 > 4 q t + 3 q t + 4 q t + 3 q t + 3 q t + q t + 3 q t + 13 6 15 6 17 7 > q t + q t + q t
In[21]:=	ColouredJones[Knot[10, 12], 2][q]
Out[21]=	-7 2 4 7 -2 12 2 3 4 5 -15 + q - -- + -- - -- + q + -- - 2 q + 27 q - 25 q - 11 q + 44 q - 6 4 3 q q q q 6 7 8 9 10 11 12 13 > 30 q - 20 q + 52 q - 28 q - 24 q + 46 q - 18 q - 23 q + 14 15 16 17 18 19 20 22 23 > 31 q - 7 q - 16 q + 15 q - q - 7 q + 5 q - 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1012

1011 1013