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

109 1011

KnotPlot

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KnotPlot

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 14 18 16 2 20 10 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:

3t-2 - 11t-1 + 17 - 11t + 3t2

Conway Polynomial:

1 + z2 + 3z4

Other knots with the same Alexander/Conway Polynomial:

{10164, ...}

Determinant and Signature:

{45, 0}

Jones Polynomial:

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

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

{...}

A2 (sl(3)) Invariant:

- q-10 + q-8 + q-6 - q-4 + 2q-2 - q6 + q8 + q12 + 2q14 - q16 - q22

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- 3a-7z + 7a-7z3 - 5a-7z5 + a-7z7 + a-6 - 8a-6z2 + 15a-6z4 - 10a-6z6 + 2a-6z8 - 6a-5z + 17a-5z3 - 10a-5z5 - a-5z7 + a-5z9 + 2a-4 - 12a-4z2 + 26a-4z4 - 21a-4z6 + 5a-4z8 - 4a-3z + 17a-3z3 - 16a-3z5 + 2a-3z7 + a-3z9 + a-2 - 4a-2z2 + 5a-2z4 - 7a-2z6 + 3a-2z8 - a-1z + 3a-1z3 - 7a-1z5 + 4a-1z7 + 1 - 2z2 - 3z4 + 4z6 - 3az3 + 4az5 - 2a2z2 + 3a2z4 + a3z3

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

{1, 2}

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 1010. 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 = 15                     1 j = 13                   1   j = 11                 2 1   j = 9               3 1     j = 7             3 2       j = 5           4 3         j = 3         3 3           j = 1       3 4             j = -1     2 4               j = -3   1 2                 j = -5   2                   j = -7 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-9 - 3q-8 + q-7 + 6q-6 - 10q-5 + 4q-4 + 10q-3 - 19q-2 + 10q-1 + 14 - 28q + 14q2 + 19q3 - 34q4 + 12q5 + 24q6 - 33q7 + 6q8 + 25q9 - 26q10 - q11 + 22q12 - 16q13 - 5q14 + 15q15 - 6q16 - 5q17 + 6q18 - q19 - 2q20 + q21
3	- q-18 + 3q-17 - q-16 - 3q-15 - 2q-14 + 7q-13 + 4q-12 - 11q-11 - 4q-10 + 9q-9 + 10q-8 - 13q-7 - 6q-6 + 4q-5 + 11q-4 - 3q-3 - 3q-2 - 7q-1 + 2 + 8q + 9q2 - 13q3 - 12q4 + 8q5 + 17q6 - 2q7 - 17q8 - 8q9 + 17q10 + 15q11 - 11q12 - 27q13 + 9q14 + 32q15 - 41q17 - 3q18 + 42q19 + 13q20 - 45q21 - 19q22 + 40q23 + 28q24 - 36q25 - 31q26 + 25q27 + 33q28 - 15q29 - 30q30 + 7q31 + 23q32 + q33 - 17q34 - 3q35 + 9q36 + 4q37 - 5q38 - 2q39 + q40 + 2q41 - q42
4	q-30 - 3q-29 + q-28 + 3q-27 - q-26 + 5q-25 - 15q-24 + 3q-23 + 9q-22 + 2q-21 + 19q-20 - 44q-19 - q-18 + 16q-17 + 17q-16 + 49q-15 - 85q-14 - 25q-13 + 11q-12 + 52q-11 + 110q-10 - 125q-9 - 80q-8 - 22q-7 + 96q-6 + 208q-5 - 141q-4 - 153q-3 - 95q-2 + 122q-1 + 324 - 119q - 201q2 - 183q3 + 101q4 + 404q5 - 72q6 - 195q7 - 236q8 + 50q9 + 416q10 - 44q11 - 148q12 - 227q13 + 4q14 + 374q15 - 45q16 - 95q17 - 183q18 - 24q19 + 309q20 - 58q21 - 44q22 - 127q23 - 45q24 + 234q25 - 71q26 + 3q27 - 65q28 - 53q29 + 156q30 - 96q31 + 31q32 - 33q34 + 97q35 - 126q36 + 20q37 + 38q38 + 10q39 + 80q40 - 135q41 - 20q42 + 30q43 + 41q44 + 92q45 - 101q46 - 47q47 - 7q48 + 33q49 + 93q50 - 44q51 - 37q52 - 30q53 + 3q54 + 63q55 - 5q56 - 11q57 - 23q58 - 12q59 + 26q60 + 3q61 + 2q62 - 7q63 - 8q64 + 6q65 + q66 + 2q67 - q68 - 2q69 + q70

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, 10]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 12, 4, 13], X[13, 1, 14, 20], X[5, 15, 6, 14], > X[19, 7, 20, 6], X[7, 19, 8, 18], X[9, 17, 10, 16], X[15, 11, 16, 10], > X[17, 9, 18, 8], X[11, 2, 12, 3]]
In[3]:=	GaussCode[Knot[10, 10]]
Out[3]=	GaussCode[-1, 10, -2, 1, -4, 5, -6, 9, -7, 8, -10, 2, -3, 4, -8, 7, -9, 6, -5, > 3]
In[4]:=	DTCode[Knot[10, 10]]
Out[4]=	DTCode[4, 12, 14, 18, 16, 2, 20, 10, 8, 6]
In[5]:=	br = BR[Knot[10, 10]]
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, 10]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 10]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 10]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 2, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 10]][t]
Out[10]=	3 11 2 17 + -- - -- - 11 t + 3 t 2 t t
In[11]:=	Conway[Knot[10, 10]][z]
Out[11]=	2 4 1 + z + 3 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 10], Knot[10, 164]}
In[13]:=	{KnotDet[Knot[10, 10]], KnotSignature[Knot[10, 10]]}
Out[13]=	{45, 0}
In[14]:=	Jones[Knot[10, 10]][q]
Out[14]=	-3 3 4 2 3 4 5 6 7 6 - q + -- - - - 7 q + 7 q - 6 q + 5 q - 3 q + 2 q - q 2 q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 10]}
In[16]:=	A2Invariant[Knot[10, 10]][q]
Out[16]=	-10 -8 -6 -4 2 6 8 12 14 16 22 -q + q + q - q + -- - q + q + q + 2 q - q - q 2 q
In[17]:=	HOMFLYPT[Knot[10, 10]][a, z]
Out[17]=	2 2 4 4 -6 2 -2 2 z 2 z 2 2 4 z z 1 - a + -- - a + z - -- + ---- - a z + z + -- + -- 4 6 4 4 2 a a a a a
In[18]:=	Kauffman[Knot[10, 10]][a, z]
Out[18]=	2 2 2 -6 2 -2 3 z 6 z 4 z z 2 8 z 12 z 4 z 1 + a + -- + a - --- - --- - --- - - - 2 z - ---- - ----- - ---- - 4 7 5 3 a 6 4 2 a a a a a a a 3 3 3 3 4 2 2 7 z 17 z 17 z 3 z 3 3 3 4 15 z > 2 a z + ---- + ----- + ----- + ---- - 3 a z + a z - 3 z + ----- + 7 5 3 a 6 a a a a 4 4 5 5 5 5 26 z 5 z 2 4 5 z 10 z 16 z 7 z 5 6 > ----- + ---- + 3 a z - ---- - ----- - ----- - ---- + 4 a z + 4 z - 4 2 7 5 3 a a a a a a 6 6 6 7 7 7 7 8 8 8 9 9 10 z 21 z 7 z z z 2 z 4 z 2 z 5 z 3 z z z > ----- - ----- - ---- + -- - -- + ---- + ---- + ---- + ---- + ---- + -- + -- 6 4 2 7 5 3 a 6 4 2 5 3 a a a a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 10]], Vassiliev[3][Knot[10, 10]]}
Out[19]=	{1, 2}
In[20]:=	Kh[Knot[10, 10]][q, t]
Out[20]=	4 1 2 1 2 2 3 3 2 - + 3 q + ----- + ----- + ----- + ---- + --- + 4 q t + 3 q t + 3 q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 5 2 5 3 7 3 7 4 9 4 9 5 11 5 > 4 q t + 3 q t + 3 q t + 2 q t + 3 q t + q t + 2 q t + 11 6 13 6 15 7 > q t + q t + q t
In[21]:=	ColouredJones[Knot[10, 10], 2][q]
Out[21]=	-9 3 -7 6 10 4 10 19 10 2 3 14 + q - -- + q + -- - -- + -- + -- - -- + -- - 28 q + 14 q + 19 q - 8 6 5 4 3 2 q q q q q q q 4 5 6 7 8 9 10 11 12 > 34 q + 12 q + 24 q - 33 q + 6 q + 25 q - 26 q - q + 22 q - 13 14 15 16 17 18 19 20 21 > 16 q - 5 q + 15 q - 6 q - 5 q + 6 q - q - 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1010

109 1011