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

K11a147 K11a149

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 16 20 12 2 18 6 22 8 14

Alexander Polynomial:

- 7t-2 + 29t-1 - 43 + 29t - 7t2

Conway Polynomial:

1 + z2 - 7z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{115, 2}

Jones Polynomial:

q-1 - 3 + 7q - 11q2 + 16q3 - 18q4 + 18q5 - 16q6 + 12q7 - 8q8 + 4q9 - q10

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

{...}

A2 (sl(3)) Invariant:

q-4 - q-2 - 1 + 4q2 - 2q4 + q6 + 4q8 - 2q10 + 2q12 - 2q14 - q16 - 4q20 + 4q22 - q26 + 3q28 - q30 - q32

HOMFLY-PT Polynomial:

- a-10 + 3a-8 + 4a-8z2 - 3a-6 - 4a-6z2 - 3a-6z4 - 2a-4z2 - 3a-4z4 + 2a-2 + 2a-2z2 - a-2z4 + z2

Kauffman Polynomial:

- a-11z + 3a-11z3 - 3a-11z5 + a-11z7 + a-10 - 4a-10z2 + 14a-10z4 - 14a-10z6 + 4a-10z8 - 2a-9z + 7a-9z3 + 5a-9z5 - 14a-9z7 + 5a-9z9 + 3a-8 - 17a-8z2 + 42a-8z4 - 37a-8z6 + 6a-8z8 + 2a-8z10 - 2a-7z + a-7z3 + 14a-7z5 - 28a-7z7 + 11a-7z9 + 3a-6 - 16a-6z2 + 35a-6z4 - 38a-6z6 + 10a-6z8 + 2a-6z10 - a-5z + 5a-5z3 - 7a-5z5 - 5a-5z7 + 6a-5z9 + 4a-4z2 - a-4z4 - 9a-4z6 + 8a-4z8 + 6a-3z3 - 10a-3z5 + 8a-3z7 - 2a-2 + 6a-2z2 - 7a-2z4 + 6a-2z6 - 2a-1z3 + 3a-1z5 - z2 + z4

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

{1, 2}

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 11148. 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 r = 6 r = 7 r = 8 r = 9 j = 21                       1 j = 19                     3   j = 17                   5 1   j = 15                 7 3     j = 13               9 5       j = 11             9 7         j = 9           9 9           j = 7         7 9             j = 5       4 9               j = 3     3 7                 j = 1   1 5                   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, Alternating, 148]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 148]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[16, 6, 17, 5], X[20, 8, 21, 7], > X[12, 10, 13, 9], X[2, 11, 3, 12], X[18, 14, 19, 13], X[6, 16, 7, 15], > X[22, 18, 1, 17], X[8, 20, 9, 19], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 148]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -8, 4, -10, 5, -2, 6, -5, 7, -11, 8, -3, 9, -7, 10, > -4, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 148]]
Out[5]=	DTCode[4, 10, 16, 20, 12, 2, 18, 6, 22, 8, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 148]][t]
Out[6]=	7 29 2 -43 - -- + -- + 29 t - 7 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 148]][z]
Out[7]=	2 4 1 + z - 7 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 148]}
In[9]:=	{KnotDet[Knot[11, Alternating, 148]], KnotSignature[Knot[11, Alternating, 148]]}
Out[9]=	{115, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 148]][q]
Out[10]=	1 2 3 4 5 6 7 8 9 10 -3 + - + 7 q - 11 q + 16 q - 18 q + 18 q - 16 q + 12 q - 8 q + 4 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 148]}
In[12]:=	A2Invariant[Knot[11, Alternating, 148]][q]
Out[12]=	-4 -2 2 4 6 8 10 12 14 16 -1 + q - q + 4 q - 2 q + q + 4 q - 2 q + 2 q - 2 q - q - 20 22 26 28 30 32 > 4 q + 4 q - q + 3 q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 148]][a, z]
Out[13]=	2 2 2 2 4 4 4 -10 3 3 2 2 4 z 4 z 2 z 2 z 3 z 3 z z -a + -- - -- + -- + z + ---- - ---- - ---- + ---- - ---- - ---- - -- 8 6 2 8 6 4 2 6 4 2 a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 148]][a, z]
Out[14]=	2 2 2 2 -10 3 3 2 z 2 z 2 z z 2 4 z 17 z 16 z 4 z a + -- + -- - -- - --- - --- - --- - -- - z - ---- - ----- - ----- + ---- + 8 6 2 11 9 7 5 10 8 6 4 a a a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 6 z 3 z 7 z z 5 z 6 z 2 z 4 14 z 42 z 35 z > ---- + ---- + ---- + -- + ---- + ---- - ---- + z + ----- + ----- + ----- - 2 11 9 7 5 3 a 10 8 6 a a a a a a a a a 4 4 5 5 5 5 5 5 6 6 z 7 z 3 z 5 z 14 z 7 z 10 z 3 z 14 z 37 z > -- - ---- - ---- + ---- + ----- - ---- - ----- + ---- - ----- - ----- - 4 2 11 9 7 5 3 a 10 8 a a a a a a a a a 6 6 6 7 7 7 7 7 8 8 38 z 9 z 6 z z 14 z 28 z 5 z 8 z 4 z 6 z > ----- - ---- + ---- + --- - ----- - ----- - ---- + ---- + ---- + ---- + 6 4 2 11 9 7 5 3 10 8 a a a a a a a a a a 8 8 9 9 9 10 10 10 z 8 z 5 z 11 z 6 z 2 z 2 z > ----- + ---- + ---- + ----- + ---- + ----- + ----- 6 4 9 7 5 8 6 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 148]], Vassiliev[3][Knot[11, Alternating, 148]]}
Out[15]=	{1, 2}
In[16]:=	Kh[Knot[11, Alternating, 148]][q, t]
Out[16]=	3 1 2 q 3 5 5 2 7 2 7 3 5 q + 3 q + ----- + --- + - + 7 q t + 4 q t + 9 q t + 7 q t + 9 q t + 3 2 q t t q t 9 3 9 4 11 4 11 5 13 5 13 6 15 6 > 9 q t + 9 q t + 9 q t + 7 q t + 9 q t + 5 q t + 7 q t + 15 7 17 7 17 8 19 8 21 9 > 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a148

K11a147 K11a149