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

K11n146 K11n148

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X14,4,15,3 X5,11,6,10 X7,20,8,21 X9,1,10,22 X11,19,12,18 X2,14,3,13 X15,9,16,8 X17,6,18,7 X19,13,20,12 X21,16,22,17

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 14 -10 -20 -22 -18 2 -8 -6 -12 -16

Alexander Polynomial:

t-4 - 4t-3 + 7t-2 - 5t-1 + 3 - 5t + 7t2 - 4t3 + t4

Conway Polynomial:

1 + 3z2 + 3z4 + 4z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{37, 4}

Jones Polynomial:

- q-1 + 3 - 4q + 6q2 - 6q3 + 6q4 - 5q5 + 4q6 - 2q7

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

{...}

A2 (sl(3)) Invariant:

- q-2 + 1 + q4 + q6 + q8 + 2q10 - 2q12 + 2q14 - q16 + q18 - q22 - 2q26 + q28

HOMFLY-PT Polynomial:

a-8z2 - 2a-6 - 6a-6z2 - 5a-6z4 - a-6z6 + 3a-4 + 11a-4z2 + 12a-4z4 + 6a-4z6 + a-4z8 - 3a-2z2 - 4a-2z4 - a-2z6

Kauffman Polynomial:

a-9z + a-8z4 - 2a-7z + 7a-7z3 - 7a-7z5 + 2a-7z7 + 2a-6 - 12a-6z2 + 24a-6z4 - 18a-6z6 + 4a-6z8 - 5a-5z + 12a-5z3 - 5a-5z5 - 5a-5z7 + 2a-5z9 + 3a-4 - 19a-4z2 + 41a-4z4 - 32a-4z6 + 7a-4z8 - 3a-3z + 9a-3z3 - 2a-3z5 - 6a-3z7 + 2a-3z9 - 7a-2z2 + 18a-2z4 - 14a-2z6 + 3a-2z8 - a-1z + 4a-1z3 - 4a-1z5 + a-1z7

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=4 is the signature of 11147. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 15                 2 j = 13               2   j = 11             3 2   j = 9           3 2     j = 7         3 3       j = 5       3 3         j = 3     2 4           j = 1   1 2             j = -1   2               j = -3 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, NonAlternating, 147]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 147]]
Out[3]=	PD[X[4, 2, 5, 1], X[14, 4, 15, 3], X[5, 11, 6, 10], X[7, 20, 8, 21], > X[9, 1, 10, 22], X[11, 19, 12, 18], X[2, 14, 3, 13], X[15, 9, 16, 8], > X[17, 6, 18, 7], X[19, 13, 20, 12], X[21, 16, 22, 17]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 147]]
Out[4]=	GaussCode[1, -7, 2, -1, -3, 9, -4, 8, -5, 3, -6, 10, 7, -2, -8, 11, -9, 6, -10, > 4, -11, 5]
In[5]:=	DTCode[Knot[11, NonAlternating, 147]]
Out[5]=	DTCode[4, 14, -10, -20, -22, -18, 2, -8, -6, -12, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 147]][t]
Out[6]=	-4 4 7 5 2 3 4 3 + t - -- + -- - - - 5 t + 7 t - 4 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 147]][z]
Out[7]=	2 4 6 8 1 + 3 z + 3 z + 4 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 147]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 147]], KnotSignature[Knot[11, NonAlternating, 147]]}
Out[9]=	{37, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 147]][q]
Out[10]=	1 2 3 4 5 6 7 3 - - - 4 q + 6 q - 6 q + 6 q - 5 q + 4 q - 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 147]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 147]][q]
Out[12]=	-2 4 6 8 10 12 14 16 18 22 26 28 1 - q + q + q + q + 2 q - 2 q + 2 q - q + q - q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 147]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 6 6 8 -2 3 z 6 z 11 z 3 z 5 z 12 z 4 z z 6 z z z -- + -- + -- - ---- + ----- - ---- - ---- + ----- - ---- - -- + ---- - -- + -- 6 4 8 6 4 2 6 4 2 6 4 2 4 a a a a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 147]][a, z]
Out[14]=	2 2 2 3 3 2 3 z 2 z 5 z 3 z z 12 z 19 z 7 z 7 z 12 z -- + -- + -- - --- - --- - --- - - - ----- - ----- - ---- + ---- + ----- + 6 4 9 7 5 3 a 6 4 2 7 5 a a a a a a a a a a a 3 3 4 4 4 4 5 5 5 5 9 z 4 z z 24 z 41 z 18 z 7 z 5 z 2 z 4 z > ---- + ---- + -- + ----- + ----- + ----- - ---- - ---- - ---- - ---- - 3 a 8 6 4 2 7 5 3 a a a a a a a a a 6 6 6 7 7 7 7 8 8 8 18 z 32 z 14 z 2 z 5 z 6 z z 4 z 7 z 3 z > ----- - ----- - ----- + ---- - ---- - ---- + -- + ---- + ---- + ---- + 6 4 2 7 5 3 a 6 4 2 a a a a a a a a a 9 9 2 z 2 z > ---- + ---- 5 3 a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 147]], Vassiliev[3][Knot[11, NonAlternating, 147]]}
Out[15]=	{3, 5}
In[16]:=	Kh[Knot[11, NonAlternating, 147]][q, t]
Out[16]=	3 3 5 1 2 q 2 q 2 q 5 7 7 2 4 q + 3 q + ----- + ---- + -- + --- + ---- + 3 q t + 3 q t + 3 q t + 3 3 2 2 t t q t q t t 9 2 9 3 11 3 11 4 13 4 15 5 > 3 q t + 2 q t + 3 q t + 2 q t + 2 q t + 2 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n147

K11n146 K11n148