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

1030 1032

KnotPlot

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KnotPlot

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 16 18 14 2 10 20 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:

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

Conway Polynomial:

1 + 2z2 + 4z4

Other knots with the same Alexander/Conway Polynomial:

{1068, ...}

Determinant and Signature:

{57, 0}

Jones Polynomial:

- q-5 + 3q-4 - 5q-3 + 7q-2 - 9q-1 + 10 - 8q + 7q2 - 4q3 + 2q4 - q5

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

{...}

A2 (sl(3)) Invariant:

- q-16 + q-14 + q-12 - 2q-10 + q-8 - q-6 - q-4 + 2q-2 + 3q2 + q6 + 2q8 - 2q10 - q16

HOMFLY-PT Polynomial:

- a-4 - a-4z2 + a-2 + a-2z2 + a-2z4 + 2 + 3z2 + 2z4 - a2 + a2z4 - a4z2

Kauffman Polynomial:

2a-5z - 3a-5z3 + a-5z5 - a-4 + 3a-4z2 - 5a-4z4 + 2a-4z6 + 2a-3z - 3a-3z3 - 2a-3z5 + 2a-3z7 - a-2 - 2a-2z2 + 3a-2z4 - 3a-2z6 + 2a-2z8 - 2a-1z + 6a-1z3 - 4a-1z5 + a-1z7 + a-1z9 + 2 - 10z2 + 20z4 - 14z6 + 5z8 - 4az + 15az3 - 12az5 + 3az7 + az9 + a2 - 3a2z2 + 5a2z4 - 6a2z6 + 3a2z8 - 2a3z + 7a3z3 - 10a3z5 + 4a3z7 + 2a4z2 - 7a4z4 + 3a4z6 - 2a5z3 + a5z5

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

{2, 1}

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 1031. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-15 - 3q-14 + q-13 + 8q-12 - 13q-11 - q-10 + 25q-9 - 26q-8 - 12q-7 + 50q-6 - 34q-5 - 30q-4 + 72q-3 - 35q-2 - 45q-1 + 78 - 27q - 47q2 + 63q3 - 13q4 - 37q5 + 38q6 - 3q7 - 21q8 + 16q9 - 8q11 + 5q12 - 2q14 + q15
3	- q-30 + 3q-29 - q-28 - 4q-27 - q-26 + 10q-25 + 3q-24 - 19q-23 - 7q-22 + 30q-21 + 17q-20 - 42q-19 - 36q-18 + 53q-17 + 64q-16 - 59q-15 - 98q-14 + 53q-13 + 138q-12 - 39q-11 - 177q-10 + 16q-9 + 212q-8 + 12q-7 - 238q-6 - 42q-5 + 256q-4 + 67q-3 - 256q-2 - 97q-1 + 258 + 105q - 227q2 - 129q3 + 208q4 + 124q5 - 160q6 - 132q7 + 128q8 + 113q9 - 79q10 - 103q11 + 52q12 + 76q13 - 24q14 - 55q15 + 10q16 + 36q17 - 6q18 - 19q19 + q20 + 13q21 - 4q22 - 5q23 + q24 + 5q25 - 3q26 - q27 + 2q29 - q30
4	q-50 - 3q-49 + q-48 + 4q-47 - 3q-46 + 4q-45 - 13q-44 + 5q-43 + 16q-42 - 12q-41 + 13q-40 - 42q-39 + 10q-38 + 52q-37 - 18q-36 + 28q-35 - 111q-34 - 5q-33 + 110q-32 + 12q-31 + 96q-30 - 220q-29 - 102q-28 + 123q-27 + 87q-26 + 296q-25 - 283q-24 - 281q-23 - 18q-22 + 111q-21 + 632q-20 - 187q-19 - 438q-18 - 316q-17 - 17q-16 + 980q-15 + 63q-14 - 466q-13 - 645q-12 - 275q-11 + 1215q-10 + 343q-9 - 371q-8 - 875q-7 - 548q-6 + 1288q-5 + 552q-4 - 215q-3 - 969q-2 - 751q-1 + 1213 + 660q - 36q2 - 923q3 - 867q4 + 987q5 + 664q6 + 170q7 - 735q8 - 892q9 + 640q10 + 542q11 + 349q12 - 427q13 - 781q14 + 274q15 + 306q16 + 403q17 - 113q18 - 536q19 + 39q20 + 63q21 + 301q22 + 65q23 - 267q24 - 21q25 - 64q26 + 146q27 + 84q28 - 95q29 + 6q30 - 70q31 + 44q32 + 41q33 - 30q34 + 23q35 - 34q36 + 9q37 + 10q38 - 14q39 + 17q40 - 9q41 + 2q42 + q43 - 7q44 + 6q45 - q46 + q47 - 2q49 + q50

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, 31]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 12, 4, 13], X[9, 14, 10, 15], X[13, 10, 14, 11], > X[15, 1, 16, 20], X[5, 17, 6, 16], X[19, 7, 20, 6], X[7, 19, 8, 18], > X[17, 9, 18, 8], X[11, 2, 12, 3]]
In[3]:=	GaussCode[Knot[10, 31]]
Out[3]=	GaussCode[-1, 10, -2, 1, -6, 7, -8, 9, -3, 4, -10, 2, -4, 3, -5, 6, -9, 8, -7, > 5]
In[4]:=	DTCode[Knot[10, 31]]
Out[4]=	DTCode[4, 12, 16, 18, 14, 2, 10, 20, 8, 6]
In[5]:=	br = BR[Knot[10, 31]]
Out[5]=	BR[5, {-1, -1, -1, -2, 1, 3, -2, 3, 3, 4, -3, 4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 31]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 31]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 31]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 2, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 31]][t]
Out[10]=	4 14 2 21 + -- - -- - 14 t + 4 t 2 t t
In[11]:=	Conway[Knot[10, 31]][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, 31]], KnotSignature[Knot[10, 31]]}
Out[13]=	{57, 0}
In[14]:=	Jones[Knot[10, 31]][q]
Out[14]=	-5 3 5 7 9 2 3 4 5 10 - q + -- - -- + -- - - - 8 q + 7 q - 4 q + 2 q - q 4 3 2 q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 31]}
In[16]:=	A2Invariant[Knot[10, 31]][q]
Out[16]=	-16 -14 -12 2 -8 -6 -4 2 2 6 8 10 -q + q + q - --- + q - q - q + -- + 3 q + q + 2 q - 2 q - 10 2 q q 16 > q
In[17]:=	HOMFLYPT[Knot[10, 31]][a, z]
Out[17]=	2 2 4 -4 -2 2 2 z z 4 2 4 z 2 4 2 - a + a - a + 3 z - -- + -- - a z + 2 z + -- + a z 4 2 2 a a a
In[18]:=	Kauffman[Knot[10, 31]][a, z]
Out[18]=	2 2 -4 -2 2 2 z 2 z 2 z 3 2 3 z 2 z 2 - a - a + a + --- + --- - --- - 4 a z - 2 a z - 10 z + ---- - ---- - 5 3 a 4 2 a a a a 3 3 3 2 2 4 2 3 z 3 z 6 z 3 3 3 5 3 > 3 a z + 2 a z - ---- - ---- + ---- + 15 a z + 7 a z - 2 a z + 5 3 a a a 4 4 5 5 5 4 5 z 3 z 2 4 4 4 z 2 z 4 z 5 > 20 z - ---- + ---- + 5 a z - 7 a z + -- - ---- - ---- - 12 a z - 4 2 5 3 a a a a a 6 6 7 7 3 5 5 5 6 2 z 3 z 2 6 4 6 2 z z > 10 a z + a z - 14 z + ---- - ---- - 6 a z + 3 a z + ---- + -- + 4 2 3 a a a a 8 9 7 3 7 8 2 z 2 8 z 9 > 3 a z + 4 a z + 5 z + ---- + 3 a z + -- + a z 2 a a
In[19]:=	{Vassiliev[2][Knot[10, 31]], Vassiliev[3][Knot[10, 31]]}
Out[19]=	{2, 1}
In[20]:=	Kh[Knot[10, 31]][q, t]
Out[20]=	5 1 2 1 3 2 4 3 5 4 - + 6 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 9 4 > 4 q t + 4 q t + 3 q t + 4 q t + q t + 3 q t + q t + q t + 11 5 > q t
In[21]:=	ColouredJones[Knot[10, 31], 2][q]
Out[21]=	-15 3 -13 8 13 -10 25 26 12 50 34 30 72 78 + q - --- + q + --- - --- - q + -- - -- - -- + -- - -- - -- + -- - 14 12 11 9 8 7 6 5 4 3 q q q q q q q q q q 35 45 2 3 4 5 6 7 8 > -- - -- - 27 q - 47 q + 63 q - 13 q - 37 q + 38 q - 3 q - 21 q + 2 q q 9 11 12 14 15 > 16 q - 8 q + 5 q - 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1031

1030 1032