© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a146 K11a148

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X16,6,17,5 X20,8,21,7 X2,10,3,9 X22,11,1,12 X18,13,19,14 X8,16,9,15 X12,17,13,18 X6,20,7,19 X14,21,15,22

Gauss Code:

{1, -5, 2, -1, 3, -10, 4, -8, 5, -2, 6, -9, 7, -11, 8, -3, 9, -7, 10, -4, 11, -6}

DT (Dowker-Thistlethwaite) Code:

4 10 16 20 2 22 18 8 12 6 14

Alexander Polynomial:

- t-4 + 6t-3 - 17t-2 + 32t-1 - 39 + 32t - 17t2 + 6t3 - t4

Conway Polynomial:

1 + 2z2 - z4 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{151, 2}

Jones Polynomial:

q-3 - 4q-2 + 9q-1 - 15 + 21q - 24q2 + 25q3 - 21q4 + 16q5 - 10q6 + 4q7 - q8

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

{K11a322, ...}

A2 (sl(3)) Invariant:

q-8 - 2q-6 + 3q-4 - 2q-2 - 1 + 4q2 - 5q4 + 5q6 - 2q8 + 2q10 + 3q12 - 3q14 + 4q16 - 3q18 - q20 + q22 - q24

HOMFLY-PT Polynomial:

- 2a-6 - 2a-6z2 - a-6z4 + 4a-4 + 9a-4z2 + 7a-4z4 + 2a-4z6 - 2a-2 - 8a-2z2 - 10a-2z4 - 5a-2z6 - a-2z8 + 1 + 3z2 + 3z4 + z6

Kauffman Polynomial:

- a-9z3 + a-9z5 + a-8z2 - 4a-8z4 + 4a-8z6 - 4a-7z + 10a-7z3 - 14a-7z5 + 9a-7z7 + 2a-6 - 5a-6z2 + 10a-6z4 - 16a-6z6 + 11a-6z8 - 7a-5z + 24a-5z3 - 27a-5z5 + 4a-5z7 + 7a-5z9 + 4a-4 - 14a-4z2 + 34a-4z4 - 45a-4z6 + 19a-4z8 + 2a-4z10 - 5a-3z + 19a-3z3 - 15a-3z5 - 13a-3z7 + 13a-3z9 + 2a-2 - 12a-2z2 + 32a-2z4 - 41a-2z6 + 15a-2z8 + 2a-2z10 - 3a-1z + 12a-1z3 - 12a-1z5 - 4a-1z7 + 6a-1z9 + 1 - 3z2 + 10z4 - 15z6 + 7z8 - az + 6az3 - 9az5 + 4az7 + a2z2 - 2a2z4 + a2z6

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

{2, 4}

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 11147. 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 r = 5 r = 6 r = 7 j = 17                       1 j = 15                     3   j = 13                   7 1   j = 11                 9 3     j = 9               12 7       j = 7             13 9         j = 5           11 12           j = 3         10 13             j = 1       6 12               j = -1     3 9                 j = -3   1 6                   j = -5   3                     j = -7 1

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]:=	Show[DrawMorseLink[Knot[11, Alternating, 147]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 147]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[16, 6, 17, 5], X[20, 8, 21, 7], > X[2, 10, 3, 9], X[22, 11, 1, 12], X[18, 13, 19, 14], X[8, 16, 9, 15], > X[12, 17, 13, 18], X[6, 20, 7, 19], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 147]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -10, 4, -8, 5, -2, 6, -9, 7, -11, 8, -3, 9, -7, 10, > -4, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 147]]
Out[5]=	DTCode[4, 10, 16, 20, 2, 22, 18, 8, 12, 6, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 147]][t]
Out[6]=	-4 6 17 32 2 3 4 -39 - t + -- - -- + -- + 32 t - 17 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 147]][z]
Out[7]=	2 4 6 8 1 + 2 z - z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 147]}
In[9]:=	{KnotDet[Knot[11, Alternating, 147]], KnotSignature[Knot[11, Alternating, 147]]}
Out[9]=	{151, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 147]][q]
Out[10]=	-3 4 9 2 3 4 5 6 7 8 -15 + q - -- + - + 21 q - 24 q + 25 q - 21 q + 16 q - 10 q + 4 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 147], Knot[11, Alternating, 322]}
In[12]:=	A2Invariant[Knot[11, Alternating, 147]][q]
Out[12]=	-8 2 3 2 2 4 6 8 10 12 14 -1 + q - -- + -- - -- + 4 q - 5 q + 5 q - 2 q + 2 q + 3 q - 3 q + 6 4 2 q q q 16 18 20 22 24 > 4 q - 3 q - q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 147]][a, z]
Out[13]=	2 2 2 4 4 4 2 4 2 2 2 z 9 z 8 z 4 z 7 z 10 z 6 1 - -- + -- - -- + 3 z - ---- + ---- - ---- + 3 z - -- + ---- - ----- + z + 6 4 2 6 4 2 6 4 2 a a a a a a a a a 6 6 8 2 z 5 z z > ---- - ---- - -- 4 2 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 147]][a, z]
Out[14]=	2 2 2 2 4 2 4 z 7 z 5 z 3 z 2 z 5 z 14 z 1 + -- + -- + -- - --- - --- - --- - --- - a z - 3 z + -- - ---- - ----- - 6 4 2 7 5 3 a 8 6 4 a a a a a a a a a 2 3 3 3 3 3 12 z 2 2 z 10 z 24 z 19 z 12 z 3 4 > ----- + a z - -- + ----- + ----- + ----- + ----- + 6 a z + 10 z - 2 9 7 5 3 a a a a a a 4 4 4 4 5 5 5 5 4 z 10 z 34 z 32 z 2 4 z 14 z 27 z 15 z > ---- + ----- + ----- + ----- - 2 a z + -- - ----- - ----- - ----- - 8 6 4 2 9 7 5 3 a a a a a a a a 5 6 6 6 6 7 12 z 5 6 4 z 16 z 45 z 41 z 2 6 9 z > ----- - 9 a z - 15 z + ---- - ----- - ----- - ----- + a z + ---- + a 8 6 4 2 7 a a a a a 7 7 7 8 8 8 9 4 z 13 z 4 z 7 8 11 z 19 z 15 z 7 z > ---- - ----- - ---- + 4 a z + 7 z + ----- + ----- + ----- + ---- + 5 3 a 6 4 2 5 a a a a a a 9 9 10 10 13 z 6 z 2 z 2 z > ----- + ---- + ----- + ----- 3 a 4 2 a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 147]], Vassiliev[3][Knot[11, Alternating, 147]]}
Out[15]=	{2, 4}
In[16]:=	Kh[Knot[11, Alternating, 147]][q, t]
Out[16]=	3 1 3 1 6 3 9 6 q 3 12 q + 10 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 13 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 9 4 11 4 > 11 q t + 12 q t + 13 q t + 9 q t + 12 q t + 7 q t + 9 q t + 11 5 13 5 13 6 15 6 17 7 > 3 q t + 7 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a147

K11a146 K11a148