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

K11a366 K11n1

Knotscape

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DrawMorseLink

PD Presentation:

X12,2,13,1 X14,4,15,3 X16,6,17,5 X18,8,19,7 X20,10,21,9 X22,12,1,11 X2,14,3,13 X4,16,5,15 X6,18,7,17 X8,20,9,19 X10,22,11,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

12 14 16 18 20 22 2 4 6 8 10

Alexander Polynomial:

t-5 - t-4 + t-3 - t-2 + t-1 - 1 + t - t2 + t3 - t4 + t5

Conway Polynomial:

1 + 15z2 + 35z4 + 28z6 + 9z8 + z10

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{11, 10}

Jones Polynomial:

q5 + q7 - q8 + q9 - q10 + q11 - q12 + q13 - q14 + q15 - q16

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

{...}

A2 (sl(3)) Invariant:

q18 + q20 + 2q22 + q24 + q26 - q42 - q44 - q46

HOMFLY-PT Polynomial:

- 5a-12 - 20a-12z2 - 21a-12z4 - 8a-12z6 - a-12z8 + 6a-10 + 35a-10z2 + 56a-10z4 + 36a-10z6 + 10a-10z8 + a-10z10

Kauffman Polynomial:

a-21z + a-20z2 - a-19z + a-19z3 - 2a-18z2 + a-18z4 + a-17z - 3a-17z3 + a-17z5 + 3a-16z2 - 4a-16z4 + a-16z6 - a-15z + 6a-15z3 - 5a-15z5 + a-15z7 - 4a-14z2 + 10a-14z4 - 6a-14z6 + a-14z8 + a-13z - 10a-13z3 + 15a-13z5 - 7a-13z7 + a-13z9 - 5a-12 + 25a-12z2 - 41a-12z4 + 29a-12z6 - 9a-12z8 + a-12z10 + 5a-11z - 20a-11z3 + 21a-11z5 - 8a-11z7 + a-11z9 - 6a-10 + 35a-10z2 - 56a-10z4 + 36a-10z6 - 10a-10z8 + a-10z10

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

{15, 55}

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=10 is the signature of 11367. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 r = 10 r = 11 j = 33                       1 j = 31                         j = 29                   1 1   j = 27                         j = 25               1 1       j = 23                         j = 21           1 1           j = 19                         j = 17       1 1               j = 15                         j = 13     1                   j = 11 1                       j = 9 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, 367]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 367]]
Out[3]=	PD[X[12, 2, 13, 1], X[14, 4, 15, 3], X[16, 6, 17, 5], X[18, 8, 19, 7], > X[20, 10, 21, 9], X[22, 12, 1, 11], X[2, 14, 3, 13], X[4, 16, 5, 15], > X[6, 18, 7, 17], X[8, 20, 9, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 367]]
Out[4]=	GaussCode[1, -7, 2, -8, 3, -9, 4, -10, 5, -11, 6, -1, 7, -2, 8, -3, 9, -4, 10, > -5, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 367]]
Out[5]=	DTCode[12, 14, 16, 18, 20, 22, 2, 4, 6, 8, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 367]][t]
Out[6]=	-5 -4 -3 -2 1 2 3 4 5 -1 + t - t + t - t + - + t - t + t - t + t t
In[7]:=	Conway[Knot[11, Alternating, 367]][z]
Out[7]=	2 4 6 8 10 1 + 15 z + 35 z + 28 z + 9 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 367]}
In[9]:=	{KnotDet[Knot[11, Alternating, 367]], KnotSignature[Knot[11, Alternating, 367]]}
Out[9]=	{11, 10}
In[10]:=	J=Jones[Knot[11, Alternating, 367]][q]
Out[10]=	5 7 8 9 10 11 12 13 14 15 16 q + q - q + q - q + q - q + q - q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 367]}
In[12]:=	A2Invariant[Knot[11, Alternating, 367]][q]
Out[12]=	18 20 22 24 26 42 44 46 q + q + 2 q + q + q - q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 367]][a, z]
Out[13]=	2 2 4 4 6 6 8 8 10 -5 6 20 z 35 z 21 z 56 z 8 z 36 z z 10 z z --- + --- - ----- + ----- - ----- + ----- - ---- + ----- - --- + ----- + --- 12 10 12 10 12 10 12 10 12 10 10 a a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 367]][a, z]
Out[14]=	2 2 2 2 -5 6 z z z z z 5 z z 2 z 3 z 4 z --- - --- + --- - --- + --- - --- + --- + --- + --- - ---- + ---- - ---- + 12 10 21 19 17 15 13 11 20 18 16 14 a a a a a a a a a a a a 2 2 3 3 3 3 3 4 4 4 25 z 35 z z 3 z 6 z 10 z 20 z z 4 z 10 z > ----- + ----- + --- - ---- + ---- - ----- - ----- + --- - ---- + ----- - 12 10 19 17 15 13 11 18 16 14 a a a a a a a a a a 4 4 5 5 5 5 6 6 6 6 41 z 56 z z 5 z 15 z 21 z z 6 z 29 z 36 z > ----- - ----- + --- - ---- + ----- + ----- + --- - ---- + ----- + ----- + 12 10 17 15 13 11 16 14 12 10 a a a a a a a a a a 7 7 7 8 8 8 9 9 10 10 z 7 z 8 z z 9 z 10 z z z z z > --- - ---- - ---- + --- - ---- - ----- + --- + --- + --- + --- 15 13 11 14 12 10 13 11 12 10 a a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 367]], Vassiliev[3][Knot[11, Alternating, 367]]}
Out[15]=	{15, 55}
In[16]:=	Kh[Knot[11, Alternating, 367]][q, t]
Out[16]=	9 11 13 2 17 3 17 4 21 5 21 6 25 7 25 8 q + q + q t + q t + q t + q t + q t + q t + q t + 29 9 29 10 33 11 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a367

K11a366 K11n1