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940 942

KnotPlot

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KnotPlot

PD Presentation:

X6271 X12,8,13,7 X14,5,15,6 X10,3,11,4 X2,11,3,12 X4,15,5,16 X8,17,9,18 X16,9,17,10 X18,14,1,13

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 14 12 16 2 18 4 8

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 2 2 3 / 4 2

Alexander Polynomial:

3t-2 - 12t-1 + 19 - 12t + 3t2

Conway Polynomial:

1 + 3z4

Other knots with the same Alexander/Conway Polynomial:

{K11n83, ...}

Determinant and Signature:

{49, 0}

Jones Polynomial:

q-6 - 3q-5 + 5q-4 - 7q-3 + 8q-2 - 8q-1 + 8 - 5q + 3q2 - q3

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

{K11n4, K11n21, ...}

A2 (sl(3)) Invariant:

q-20 + q-18 - 2q-16 - q-12 - 2q-10 + 2q-8 + 2q-4 + q-2 + 2q2 - 2q4 + q6 + q8 - q10

HOMFLY-PT Polynomial:

- a-2z2 + z4 + 3a2 + 4a2z2 + 2a2z4 - 3a4 - 3a4z2 + a6

Kauffman Polynomial:

a-3z3 - a-2z2 + 3a-2z4 - 3a-1z3 + 5a-1z5 + 6z2 - 11z4 + 7z6 - 2az + 6az3 - 11az5 + 6az7 - 3a2 + 17a2z2 - 23a2z4 + 5a2z6 + 2a2z8 - 4a3z + 19a3z3 - 26a3z5 + 9a3z7 - 3a4 + 13a4z2 - 12a4z4 - a4z6 + 2a4z8 - 2a5z + 9a5z3 - 10a5z5 + 3a5z7 - a6 + 3a6z2 - 3a6z4 + a6z6

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

{0, 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 941. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 7                   1 j = 5                 2   j = 3               3 1   j = 1             5 2     j = -1           4 4       j = -3         4 4         j = -5       3 4           j = -7     2 4             j = -9   1 3               j = -11   2                 j = -13 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-18 - 3q-17 - q-16 + 11q-15 - 9q-14 - 13q-13 + 29q-12 - 7q-11 - 35q-10 + 42q-9 + 5q-8 - 55q-7 + 44q-6 + 20q-5 - 64q-4 + 36q-3 + 30q-2 - 57q-1 + 22 + 27q - 35q2 + 9q3 + 13q4 - 13q5 + 4q6 + 2q7 - 3q8 + q9
3	q-36 - 3q-35 - q-34 + 5q-33 + 9q-32 - 9q-31 - 23q-30 + 7q-29 + 41q-28 + 9q-27 - 61q-26 - 36q-25 + 71q-24 + 72q-23 - 66q-22 - 111q-21 + 45q-20 + 147q-19 - 15q-18 - 168q-17 - 26q-16 + 181q-15 + 68q-14 - 184q-13 - 108q-12 + 178q-11 + 146q-10 - 166q-9 - 177q-8 + 144q-7 + 206q-6 - 124q-5 - 215q-4 + 86q-3 + 220q-2 - 59q-1 - 190 + 18q + 165q2 - 4q3 - 113q4 - 13q5 + 74q6 + 10q7 - 37q8 - 7q9 + 19q10 - q11 - 6q12 + 2q13 + q14 - q15 - 2q16 + 3q17 - q18

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, 41]]
Out[2]=	PD[X[6, 2, 7, 1], X[12, 8, 13, 7], X[14, 5, 15, 6], X[10, 3, 11, 4], > X[2, 11, 3, 12], X[4, 15, 5, 16], X[8, 17, 9, 18], X[16, 9, 17, 10], > X[18, 14, 1, 13]]
In[3]:=	GaussCode[Knot[9, 41]]
Out[3]=	GaussCode[1, -5, 4, -6, 3, -1, 2, -7, 8, -4, 5, -2, 9, -3, 6, -8, 7, -9]
In[4]:=	DTCode[Knot[9, 41]]
Out[4]=	DTCode[6, 10, 14, 12, 16, 2, 18, 4, 8]
In[5]:=	br = BR[Knot[9, 41]]
Out[5]=	BR[5, {-1, -1, -2, 1, 3, 2, 2, -4, -3, 2, -3, -4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[9, 41]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[9, 41]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[9, 41]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 3, 4, 2}
In[10]:=	alex = Alexander[Knot[9, 41]][t]
Out[10]=	3 12 2 19 + -- - -- - 12 t + 3 t 2 t t
In[11]:=	Conway[Knot[9, 41]][z]
Out[11]=	4 1 + 3 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 41], Knot[11, NonAlternating, 83]}
In[13]:=	{KnotDet[Knot[9, 41]], KnotSignature[Knot[9, 41]]}
Out[13]=	{49, 0}
In[14]:=	Jones[Knot[9, 41]][q]
Out[14]=	-6 3 5 7 8 8 2 3 8 + q - -- + -- - -- + -- - - - 5 q + 3 q - q 5 4 3 2 q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 41], Knot[11, NonAlternating, 4], Knot[11, NonAlternating, 21]}
In[16]:=	A2Invariant[Knot[9, 41]][q]
Out[16]=	-20 -18 2 -12 2 2 2 -2 2 4 6 8 10 q + q - --- - q - --- + -- + -- + q + 2 q - 2 q + q + q - q 16 10 8 4 q q q q
In[17]:=	HOMFLYPT[Knot[9, 41]][a, z]
Out[17]=	2 2 4 6 z 2 2 4 2 4 2 4 3 a - 3 a + a - -- + 4 a z - 3 a z + z + 2 a z 2 a
In[18]:=	Kauffman[Knot[9, 41]][a, z]
Out[18]=	2 2 4 6 3 5 2 z 2 2 4 2 -3 a - 3 a - a - 2 a z - 4 a z - 2 a z + 6 z - -- + 17 a z + 13 a z + 2 a 3 3 4 6 2 z 3 z 3 3 3 5 3 4 3 z > 3 a z + -- - ---- + 6 a z + 19 a z + 9 a z - 11 z + ---- - 3 a 2 a a 5 2 4 4 4 6 4 5 z 5 3 5 5 5 > 23 a z - 12 a z - 3 a z + ---- - 11 a z - 26 a z - 10 a z + a 6 2 6 4 6 6 6 7 3 7 5 7 2 8 > 7 z + 5 a z - a z + a z + 6 a z + 9 a z + 3 a z + 2 a z + 4 8 > 2 a z
In[19]:=	{Vassiliev[2][Knot[9, 41]], Vassiliev[3][Knot[9, 41]]}
Out[19]=	{0, 1}
In[20]:=	Kh[Knot[9, 41]][q, t]
Out[20]=	4 1 2 1 3 2 4 3 4 - + 5 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 4 4 4 3 3 2 5 2 7 3 > ----- + ---- + --- + 2 q t + 3 q t + q t + 2 q t + q t 3 2 3 q t q t q t
In[21]:=	ColouredJones[Knot[9, 41], 2][q]
Out[21]=	-18 3 -16 11 9 13 29 7 35 42 5 55 22 + q - --- - q + --- - --- - --- + --- - --- - --- + -- + -- - -- + 17 15 14 13 12 11 10 9 8 7 q q q q q q q q q q 44 20 64 36 30 57 2 3 4 5 6 > -- + -- - -- + -- + -- - -- + 27 q - 35 q + 9 q + 13 q - 13 q + 4 q + 6 5 4 3 2 q q q q q q 7 8 9 > 2 q - 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 941

940 942