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

1036 1038

KnotPlot

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KnotPlot

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 16 12 2 8 20 18 6 14

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 FullyAmphicheiral 2 2 2 / NotAvailable 1

Alexander Polynomial:

4t-2 - 13t-1 + 19 - 13t + 4t2

Conway Polynomial:

1 + 3z2 + 4z4

Other knots with the same Alexander/Conway Polynomial:

{1028, ...}

Determinant and Signature:

{53, 0}

Jones Polynomial:

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

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

{...}

A2 (sl(3)) Invariant:

- q-16 - 2q-10 + 2q-8 + q-6 + 2q-2 - 1 + 2q2 + q6 + 2q8 - 2q10 - q16

HOMFLY-PT Polynomial:

- a-4 - a-4z2 + a-2 + a-2z2 + a-2z4 + 1 + 3z2 + 2z4 + a2 + a2z2 + a2z4 - a4 - 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-2z4 - 3a-2z6 + 2a-2z8 - a-1z + a-1z3 + a-1z9 + 1 - 6z2 + 14z4 - 10z6 + 4z8 - az + az3 + az9 - a2 + 2a2z4 - 3a2z6 + 2a2z8 + 2a3z - 3a3z3 - 2a3z5 + 2a3z7 - a4 + 3a4z2 - 5a4z4 + 2a4z6 + 2a5z - 3a5z3 + a5z5

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

{3, 0}

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 1037. 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           5 4         j = -1         4 5           j = -3       3 4             j = -5     1 4               j = -7   1 3                 j = -9   1                   j = -11 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-15 - 2q-14 + 5q-12 - 8q-11 + 17q-9 - 21q-8 - 6q-7 + 40q-6 - 34q-5 - 20q-4 + 63q-3 - 38q-2 - 33q-1 + 73 - 33q - 38q2 + 63q3 - 20q4 - 34q5 + 40q6 - 6q7 - 21q8 + 17q9 - 8q11 + 5q12 - 2q14 + q15
3	- q-30 + 2q-29 - q-27 - 3q-26 + 5q-25 + q-24 - 6q-23 - 4q-22 + 15q-21 + 4q-20 - 23q-19 - 14q-18 + 41q-17 + 27q-16 - 56q-15 - 53q-14 + 67q-13 + 92q-12 - 79q-11 - 128q-10 + 71q-9 + 175q-8 - 66q-7 - 206q-6 + 44q-5 + 242q-4 - 33q-3 - 251q-2 + 6q-1 + 265 + 6q - 251q2 - 33q3 + 242q4 + 44q5 - 206q6 - 66q7 + 175q8 + 71q9 - 128q10 - 79q11 + 92q12 + 67q13 - 53q14 - 56q15 + 27q16 + 41q17 - 14q18 - 23q19 + 4q20 + 15q21 - 4q22 - 6q23 + q24 + 5q25 - 3q26 - q27 + 2q29 - q30
4	q-50 - 2q-49 + q-47 - q-46 + 6q-45 - 7q-44 + q-43 + 3q-42 - 9q-41 + 15q-40 - 16q-39 + 9q-38 + 16q-37 - 27q-36 + 19q-35 - 46q-34 + 24q-33 + 63q-32 - 29q-31 + 23q-30 - 140q-29 + q-28 + 143q-27 + 42q-26 + 97q-25 - 298q-24 - 140q-23 + 166q-22 + 191q-21 + 339q-20 - 423q-19 - 401q-18 + 33q-17 + 315q-16 + 724q-15 - 403q-14 - 657q-13 - 243q-12 + 318q-11 + 1111q-10 - 256q-9 - 803q-8 - 528q-7 + 220q-6 + 1363q-5 - 78q-4 - 816q-3 - 724q-2 + 80q-1 + 1447 + 80q - 724q2 - 816q3 - 78q4 + 1363q5 + 220q6 - 528q7 - 803q8 - 256q9 + 1111q10 + 318q11 - 243q12 - 657q13 - 403q14 + 724q15 + 315q16 + 33q17 - 401q18 - 423q19 + 339q20 + 191q21 + 166q22 - 140q23 - 298q24 + 97q25 + 42q26 + 143q27 + q28 - 140q29 + 23q30 - 29q31 + 63q32 + 24q33 - 46q34 + 19q35 - 27q36 + 16q37 + 9q38 - 16q39 + 15q40 - 9q41 + 3q42 + 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, 37]]
Out[2]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[12, 8, 13, 7], X[8, 12, 9, 11], > X[18, 15, 19, 16], X[16, 5, 17, 6], X[6, 17, 7, 18], X[20, 13, 1, 14], > X[14, 19, 15, 20], X[2, 10, 3, 9]]
In[3]:=	GaussCode[Knot[10, 37]]
Out[3]=	GaussCode[1, -10, 2, -1, 6, -7, 3, -4, 10, -2, 4, -3, 8, -9, 5, -6, 7, -5, 9, > -8]
In[4]:=	DTCode[Knot[10, 37]]
Out[4]=	DTCode[4, 10, 16, 12, 2, 8, 20, 18, 6, 14]
In[5]:=	br = BR[Knot[10, 37]]
Out[5]=	BR[5, {-1, -1, -1, -2, 1, 3, -2, 3, 4, -3, 4, 4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 37]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 37]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 37]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{FullyAmphicheiral, 2, 2, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 37]][t]
Out[10]=	4 13 2 19 + -- - -- - 13 t + 4 t 2 t t
In[11]:=	Conway[Knot[10, 37]][z]
Out[11]=	2 4 1 + 3 z + 4 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 28], Knot[10, 37]}
In[13]:=	{KnotDet[Knot[10, 37]], KnotSignature[Knot[10, 37]]}
Out[13]=	{53, 0}
In[14]:=	Jones[Knot[10, 37]][q]
Out[14]=	-5 2 4 7 8 2 3 4 5 9 - 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, 37]}
In[16]:=	A2Invariant[Knot[10, 37]][q]
Out[16]=	-16 2 2 -6 2 2 6 8 10 16 -1 - q - --- + -- + q + -- + 2 q + q + 2 q - 2 q - q 10 8 2 q q q
In[17]:=	HOMFLYPT[Knot[10, 37]][a, z]
Out[17]=	2 2 4 -4 -2 2 4 2 z z 2 2 4 2 4 z 2 4 1 - a + a + a - a + 3 z - -- + -- + a z - a z + 2 z + -- + a z 4 2 2 a a a
In[18]:=	Kauffman[Knot[10, 37]][a, z]
Out[18]=	2 -4 -2 2 4 2 z 2 z z 3 5 2 3 z 1 - a - a - a - a + --- + --- - - - a z + 2 a z + 2 a z - 6 z + ---- + 5 3 a 4 a a a 3 3 3 4 4 2 3 z 3 z z 3 3 3 5 3 4 5 z > 3 a z - ---- - ---- + -- + a z - 3 a z - 3 a z + 14 z - ---- + 5 3 a 4 a a a 4 5 5 6 2 z 2 4 4 4 z 2 z 3 5 5 5 6 2 z > ---- + 2 a z - 5 a z + -- - ---- - 2 a z + a z - 10 z + ---- - 2 5 3 4 a a a a 6 7 8 9 3 z 2 6 4 6 2 z 3 7 8 2 z 2 8 z > ---- - 3 a z + 2 a z + ---- + 2 a z + 4 z + ---- + 2 a z + -- + 2 3 2 a a a a 9 > a z
In[19]:=	{Vassiliev[2][Knot[10, 37]], Vassiliev[3][Knot[10, 37]]}
Out[19]=	{3, 0}
In[20]:=	Kh[Knot[10, 37]][q, t]
Out[20]=	5 1 1 1 3 1 4 3 4 4 - + 5 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, 37], 2][q]
Out[21]=	-15 2 5 8 17 21 6 40 34 20 63 38 33 73 + q - --- + --- - --- + -- - -- - -- + -- - -- - -- + -- - -- - -- - 14 12 11 9 8 7 6 5 4 3 2 q q q q q q q q q q q q 2 3 4 5 6 7 8 9 > 33 q - 38 q + 63 q - 20 q - 34 q + 40 q - 6 q - 21 q + 17 q - 11 12 14 15 > 8 q + 5 q - 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1037

1036 1038