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947 949

KnotPlot

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KnotPlot

PD Presentation:

X1425 X12,8,13,7 X3,11,4,10 X11,3,12,2 X14,6,15,5 X6,14,7,13 X15,18,16,1 X9,17,10,16 X17,9,18,8

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

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 2 3 / 4--6 2

Alexander Polynomial:

- t-2 + 7t-1 - 11 + 7t - t2

Conway Polynomial:

1 + 3z2 - z4

Other knots with the same Alexander/Conway Polynomial:

{K11n1, ...}

Determinant and Signature:

{27, 2}

Jones Polynomial:

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

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

{...}

A2 (sl(3)) Invariant:

q-4 - q-2 - 1 + q2 - q4 + 2q6 + q8 + 2q10 + 2q12 + q16 - 2q18 - 2q20

HOMFLY-PT Polynomial:

- 2a-6 + 3a-4 + 3a-4z2 - a-2z2 - a-2z4 + z2

Kauffman Polynomial:

- 4a-7z + 3a-7z3 + 2a-6 - a-6z2 + a-6z6 - 5a-5z + 5a-5z3 - a-5z5 + a-5z7 + 3a-4 + 2a-4z2 - 6a-4z4 + 4a-4z6 - a-3z - 3a-3z3 + 2a-3z5 + a-3z7 + 2a-2z2 - 5a-2z4 + 3a-2z6 - 5a-1z3 + 3a-1z5 - z2 + z4

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

{3, 5}

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

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 13               2 j = 11             1   j = 9           3 2   j = 7         3 1     j = 5       1 3       j = 3     3 3         j = 1   1 2           j = -1   2             j = -3 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-4 - 3q-3 + 9q-1 - 9 - 6q + 20q2 - 12q3 - 15q4 + 29q5 - 10q6 - 21q7 + 32q8 - 7q9 - 21q10 + 23q11 - 2q12 - 15q13 + 10q14 + q15 - 6q16 + 2q17 + q18
3	q-9 - 3q-8 + 5q-6 + 4q-5 - 9q-4 - 13q-3 + 14q-2 + 22q-1 - 12 - 37q + 8q2 + 53q3 + 2q4 - 67q5 - 15q6 + 79q7 + 26q8 - 83q9 - 44q10 + 94q11 + 47q12 - 86q13 - 62q14 + 92q15 + 58q16 - 74q17 - 67q18 + 68q19 + 61q20 - 50q21 - 58q22 + 32q23 + 49q24 - 19q25 - 37q26 + 5q27 + 27q28 - q29 - 12q30 - 5q31 + 9q32 + q33 - 2q35
4	q-16 - 3q-15 + 5q-13 + 4q-11 - 16q-10 - 6q-9 + 14q-8 + 9q-7 + 31q-6 - 40q-5 - 39q-4 + 5q-3 + 24q-2 + 102q-1 - 41 - 90q - 58q2 + 6q3 + 213q4 + 22q5 - 119q6 - 166q7 - 70q8 + 306q9 + 127q10 - 93q11 - 268q12 - 178q13 + 350q14 + 222q15 - 36q16 - 327q17 - 269q18 + 353q19 + 278q20 + 18q21 - 342q22 - 321q23 + 324q24 + 293q25 + 66q26 - 312q27 - 336q28 + 247q29 + 273q30 + 114q31 - 238q32 - 314q33 + 136q34 + 208q35 + 145q36 - 127q37 - 242q38 + 30q39 + 109q40 + 133q41 - 25q42 - 137q43 - 23q44 + 25q45 + 76q46 + 20q47 - 46q48 - 20q49 - 9q50 + 21q51 + 14q52 - 5q53 - 4q54 - 6q55 + 2q57 + q58

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[9, 48]]
Out[2]=	PD[X[1, 4, 2, 5], X[12, 8, 13, 7], X[3, 11, 4, 10], X[11, 3, 12, 2], > X[14, 6, 15, 5], X[6, 14, 7, 13], X[15, 18, 16, 1], X[9, 17, 10, 16], > X[17, 9, 18, 8]]
In[3]:=	GaussCode[Knot[9, 48]]
Out[3]=	GaussCode[-1, 4, -3, 1, 5, -6, 2, 9, -8, 3, -4, -2, 6, -5, -7, 8, -9, 7]
In[4]:=	DTCode[Knot[9, 48]]
Out[4]=	DTCode[4, 10, -14, -12, 16, 2, -6, 18, 8]
In[5]:=	br = BR[Knot[9, 48]]
Out[5]=	BR[4, {1, 1, 2, -1, 2, 1, -3, 2, -1, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[9, 48]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[9, 48]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[9, 48]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 3, {4, 6}, 2}
In[10]:=	alex = Alexander[Knot[9, 48]][t]
Out[10]=	-2 7 2 -11 - t + - + 7 t - t t
In[11]:=	Conway[Knot[9, 48]][z]
Out[11]=	2 4 1 + 3 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 48], Knot[11, NonAlternating, 1]}
In[13]:=	{KnotDet[Knot[9, 48]], KnotSignature[Knot[9, 48]]}
Out[13]=	{27, 2}
In[14]:=	Jones[Knot[9, 48]][q]
Out[14]=	1 2 3 4 5 6 -3 + - + 4 q - 4 q + 6 q - 4 q + 3 q - 2 q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 48]}
In[16]:=	A2Invariant[Knot[9, 48]][q]
Out[16]=	-4 -2 2 4 6 8 10 12 16 18 20 -1 + q - q + q - q + 2 q + q + 2 q + 2 q + q - 2 q - 2 q
In[17]:=	HOMFLYPT[Knot[9, 48]][a, z]
Out[17]=	2 2 4 -2 3 2 3 z z z -- + -- + z + ---- - -- - -- 6 4 4 2 2 a a a a a
In[18]:=	Kauffman[Knot[9, 48]][a, z]
Out[18]=	2 2 2 3 3 3 3 2 3 4 z 5 z z 2 z 2 z 2 z 3 z 5 z 3 z 5 z -- + -- - --- - --- - -- - z - -- + ---- + ---- + ---- + ---- - ---- - ---- + 6 4 7 5 3 6 4 2 7 5 3 a a a a a a a a a a a a 4 4 5 5 5 6 6 6 7 7 4 6 z 5 z z 2 z 3 z z 4 z 3 z z z > z - ---- - ---- - -- + ---- + ---- + -- + ---- + ---- + -- + -- 4 2 5 3 a 6 4 2 5 3 a a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[9, 48]], Vassiliev[3][Knot[9, 48]]}
Out[19]=	{3, 5}
In[20]:=	Kh[Knot[9, 48]][q, t]
Out[20]=	3 1 2 q 3 5 5 2 7 2 7 3 2 q + 3 q + ----- + --- + - + 3 q t + q t + 3 q t + 3 q t + q t + 3 2 q t t q t 9 3 9 4 11 4 13 5 > 3 q t + 2 q t + q t + 2 q t
In[21]:=	ColouredJones[Knot[9, 48], 2][q]
Out[21]=	-4 3 9 2 3 4 5 6 7 -9 + q - -- + - - 6 q + 20 q - 12 q - 15 q + 29 q - 10 q - 21 q + 3 q q 8 9 10 11 12 13 14 15 16 > 32 q - 7 q - 21 q + 23 q - 2 q - 15 q + 10 q + q - 6 q + 17 18 > 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 948

947 949