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10146 10148

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KnotPlot

PD Presentation:

X4251 X10,4,11,3 X5,14,6,15 X15,20,16,1 X12,7,13,8 X8,18,9,17 X19,7,20,6 X16,12,17,11 X18,13,19,14 X2,10,3,9

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 -14 12 2 16 18 -20 8 -6

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 Chiral 1 2 3 / NotAvailable 1

Alexander Polynomial:

- 2t-2 + 7t-1 - 9 + 7t - 2t2

Conway Polynomial:

1 - z2 - 2z4

Other knots with the same Alexander/Conway Polynomial:

{811, K11n122, ...}

Determinant and Signature:

{27, 2}

Jones Polynomial:

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

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

{...}

A2 (sl(3)) Invariant:

q-10 + q-4 - q-2 + 2q6 + q10 - q12 - q14 + q16

HOMFLY-PT Polynomial:

a-4z2 + a-2 - a-2z2 - a-2z4 - 1 - 2z2 - z4 + a2 + a2z2

Kauffman Polynomial:

a-6z2 - a-5z + 3a-5z3 + a-4z6 - 3a-3z + 8a-3z3 - 6a-3z5 + 2a-3z7 - a-2 + a-2z2 - 2a-2z4 - a-2z6 + a-2z8 - 4a-1z + 13a-1z3 - 14a-1z5 + 4a-1z7 - 1 + 6z2 - 6z4 - z6 + z8 - 2az + 8az3 - 8az5 + 2az7 - a2 + 4a2z2 - 4a2z4 + a2z6

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

{-1, 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=2 is the signature of 10147. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-10 - 2q-9 - q-8 + 6q-7 - 4q-6 - 6q-5 + 12q-4 - 2q-3 - 13q-2 + 13q-1 + 3 - 17q + 11q2 + 9q3 - 17q4 + 6q5 + 12q6 - 14q7 + q8 + 8q9 - 7q10 + 3q12 - q13
3	q-21 - 2q-20 - q-19 + 2q-18 + 5q-17 - 3q-16 - 9q-15 + q-14 + 14q-13 + 3q-12 - 16q-11 - 11q-10 + 16q-9 + 17q-8 - 10q-7 - 22q-6 + 2q-5 + 23q-4 + 8q-3 - 20q-2 - 17q-1 + 14 + 26q - 9q2 - 30q3 - 2q4 + 40q5 + 6q6 - 41q7 - 16q8 + 48q9 + 19q10 - 44q11 - 28q12 + 42q13 + 29q14 - 33q15 - 30q16 + 22q17 + 27q18 - 12q19 - 18q20 + 2q21 + 13q22 - 6q24 - q25 + q26 + 2q27 - q28
4	q-36 - 2q-35 - q-34 + 2q-33 + q-32 + 6q-31 - 7q-30 - 7q-29 + q-27 + 24q-26 - 5q-25 - 15q-24 - 13q-23 - 15q-22 + 41q-21 + 12q-20 + q-19 - 17q-18 - 52q-17 + 26q-16 + 14q-15 + 35q-14 + 20q-13 - 62q-12 - 10q-11 - 33q-10 + 38q-9 + 79q-8 - 18q-7 - 16q-6 - 99q-5 - 15q-4 + 107q-3 + 52q-2 + 27q-1 - 140 - 94q + 93q2 + 109q3 + 87q4 - 150q5 - 163q6 + 64q7 + 150q8 + 135q9 - 149q10 - 213q11 + 37q12 + 181q13 + 172q14 - 141q15 - 251q16 + 2q17 + 193q18 + 200q19 - 106q20 - 255q21 - 43q22 + 153q23 + 202q24 - 36q25 - 201q26 - 74q27 + 72q28 + 151q29 + 21q30 - 105q31 - 61q32 + 5q33 + 70q34 + 30q35 - 30q36 - 23q37 - 11q38 + 16q39 + 12q40 - 4q41 - 3q42 - 4q43 + 2q44 + 2q45 - q46
5	q-55 - 2q-54 - q-53 + 2q-52 + q-51 + 2q-50 + 2q-49 - 5q-48 - 9q-47 + 5q-45 + 10q-44 + 13q-43 - q-42 - 20q-41 - 23q-40 - 6q-39 + 14q-38 + 33q-37 + 30q-36 - 3q-35 - 36q-34 - 45q-33 - 25q-32 + 15q-31 + 52q-30 + 56q-29 + 22q-28 - 30q-27 - 71q-26 - 66q-25 - 22q-24 + 47q-23 + 101q-22 + 93q-21 + 11q-20 - 92q-19 - 153q-18 - 110q-17 + 35q-16 + 185q-15 + 210q-14 + 68q-13 - 159q-12 - 293q-11 - 200q-10 + 78q-9 + 336q-8 + 337q-7 + 40q-6 - 326q-5 - 448q-4 - 189q-3 + 269q-2 + 536q-1 + 336 - 190q - 574q2 - 470q3 + 79q4 + 603q5 + 583q6 + 12q7 - 596q8 - 671q9 - 114q10 + 599q11 + 751q12 + 175q13 - 589q14 - 808q15 - 249q16 + 598q17 + 874q18 + 290q19 - 597q20 - 920q21 - 357q22 + 594q23 + 972q24 + 407q25 - 553q26 - 995q27 - 490q28 + 488q29 + 992q30 + 552q31 - 377q32 - 927q33 - 609q34 + 238q35 + 819q36 + 620q37 - 99q38 - 655q39 - 574q40 - 27q41 + 464q42 + 492q43 + 102q44 - 295q45 - 359q46 - 130q47 + 144q48 + 240q49 + 119q50 - 58q51 - 133q52 - 81q53 + 10q54 + 60q55 + 47q56 + 7q57 - 27q58 - 20q59 - 2q60 + 5q61 + 7q62 + 3q63 - 3q64 - 2q65 + q66

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, 147]]
Out[2]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[5, 14, 6, 15], X[15, 20, 16, 1], > X[12, 7, 13, 8], X[8, 18, 9, 17], X[19, 7, 20, 6], X[16, 12, 17, 11], > X[18, 13, 19, 14], X[2, 10, 3, 9]]
In[3]:=	GaussCode[Knot[10, 147]]
Out[3]=	GaussCode[1, -10, 2, -1, -3, 7, 5, -6, 10, -2, 8, -5, 9, 3, -4, -8, 6, -9, -7, > 4]
In[4]:=	DTCode[Knot[10, 147]]
Out[4]=	DTCode[4, 10, -14, 12, 2, 16, 18, -20, 8, -6]
In[5]:=	br = BR[Knot[10, 147]]
Out[5]=	BR[4, {1, 1, 1, -2, 1, -2, -3, 2, -1, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 147]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 147]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 147]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 1, 2, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 147]][t]
Out[10]=	2 7 2 -9 - -- + - + 7 t - 2 t 2 t t
In[11]:=	Conway[Knot[10, 147]][z]
Out[11]=	2 4 1 - z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[8, 11], Knot[10, 147], Knot[11, NonAlternating, 122]}
In[13]:=	{KnotDet[Knot[10, 147]], KnotSignature[Knot[10, 147]]}
Out[13]=	{27, 2}
In[14]:=	Jones[Knot[10, 147]][q]
Out[14]=	-3 2 3 2 3 4 5 -4 + q - -- + - + 5 q - 4 q + 4 q - 3 q + q 2 q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 147]}
In[16]:=	A2Invariant[Knot[10, 147]][q]
Out[16]=	-10 -4 -2 6 10 12 14 16 q + q - q + 2 q + q - q - q + q
In[17]:=	HOMFLYPT[Knot[10, 147]][a, z]
Out[17]=	2 2 4 -2 2 2 z z 2 2 4 z -1 + a + a - 2 z + -- - -- + a z - z - -- 4 2 2 a a a
In[18]:=	Kauffman[Knot[10, 147]][a, z]
Out[18]=	2 2 3 -2 2 z 3 z 4 z 2 z z 2 2 3 z -1 - a - a - -- - --- - --- - 2 a z + 6 z + -- + -- + 4 a z + ---- + 5 3 a 6 2 5 a a a a a 3 3 4 5 5 8 z 13 z 3 4 2 z 2 4 6 z 14 z 5 > ---- + ----- + 8 a z - 6 z - ---- - 4 a z - ---- - ----- - 8 a z - 3 a 2 3 a a a a 6 6 7 7 8 6 z z 2 6 2 z 4 z 7 8 z > z + -- - -- + a z + ---- + ---- + 2 a z + z + -- 4 2 3 a 2 a a a a
In[19]:=	{Vassiliev[2][Knot[10, 147]], Vassiliev[3][Knot[10, 147]]}
Out[19]=	{-1, 0}
In[20]:=	Kh[Knot[10, 147]][q, t]
Out[20]=	3 1 1 1 2 1 2 2 q 3 3 q + 3 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 2 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 11 4 > 2 q t + 2 q t + 2 q t + q t + 2 q t + q t
In[21]:=	ColouredJones[Knot[10, 147], 2][q]
Out[21]=	-10 2 -8 6 4 6 12 2 13 13 2 3 3 + q - -- - q + -- - -- - -- + -- - -- - -- + -- - 17 q + 11 q + 9 q - 9 7 6 5 4 3 2 q q q q q q q q 4 5 6 7 8 9 10 12 13 > 17 q + 6 q + 12 q - 14 q + q + 8 q - 7 q + 3 q - q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10147

10146 10148