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

K11a246 K11a248

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 14 22 20 18 16 2 12 10 8 6

Alexander Polynomial:

5t-1 - 9 + 5t

Conway Polynomial:

1 + 5z2

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{19, 2}

Jones Polynomial:

q - q2 + 2q3 - 2q4 + 2q5 - 2q6 + 2q7 - 2q8 + 2q9 - q10 + q11 - q12

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

{...}

A2 (sl(3)) Invariant:

q2 + q6 + q8 + q28 + q30 - q36 - q38

HOMFLY-PT Polynomial:

- a-12 + a-10 + a-10z2 + a-8z2 + a-6z2 + a-4z2 + a-2 + a-2z2

Kauffman Polynomial:

5a-13z - 20a-13z3 + 21a-13z5 - 8a-13z7 + a-13z9 - a-12 + 11a-12z2 - 25a-12z4 + 22a-12z6 - 8a-12z8 + a-12z10 + 5a-11z - 24a-11z3 + 31a-11z5 - 14a-11z7 + 2a-11z9 - a-10 + 10a-10z2 - 19a-10z4 + 17a-10z6 - 7a-10z8 + a-10z10 - a-9z3 + 6a-9z5 - 5a-9z7 + a-9z9 + 3a-8z4 - 4a-8z6 + a-8z8 + a-7z3 - 3a-7z5 + a-7z7 - 2a-6z4 + a-6z6 - a-5z3 + a-5z5 + a-4z4 + a-3z3 - a-2 + a-2z2

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

{5, 15}

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 11247. 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 = 25                       1 j = 23                         j = 21                   1 1   j = 19                 1       j = 17               1 1       j = 15             1 1         j = 13           1 1           j = 11         1 1             j = 9       1 1               j = 7     1 1                 j = 5     1                   j = 3 1 1                     j = 1 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, 247]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 247]]
Out[3]=	PD[X[4, 2, 5, 1], X[14, 4, 15, 3], X[22, 6, 1, 5], X[20, 8, 21, 7], > X[18, 10, 19, 9], X[16, 12, 17, 11], X[2, 14, 3, 13], X[12, 16, 13, 15], > X[10, 18, 11, 17], X[8, 20, 9, 19], X[6, 22, 7, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 247]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -11, 4, -10, 5, -9, 6, -8, 7, -2, 8, -6, 9, -5, 10, > -4, 11, -3]
In[5]:=	DTCode[Knot[11, Alternating, 247]]
Out[5]=	DTCode[4, 14, 22, 20, 18, 16, 2, 12, 10, 8, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 247]][t]
Out[6]=	5 -9 + - + 5 t t
In[7]:=	Conway[Knot[11, Alternating, 247]][z]
Out[7]=	2 1 + 5 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 247]}
In[9]:=	{KnotDet[Knot[11, Alternating, 247]], KnotSignature[Knot[11, Alternating, 247]]}
Out[9]=	{19, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 247]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 12 q - q + 2 q - 2 q + 2 q - 2 q + 2 q - 2 q + 2 q - q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 247]}
In[12]:=	A2Invariant[Knot[11, Alternating, 247]][q]
Out[12]=	2 6 8 28 30 36 38 q + q + q + q + q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 247]][a, z]
Out[13]=	2 2 2 2 2 -12 -10 -2 z z z z z -a + a + a + --- + -- + -- + -- + -- 10 8 6 4 2 a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 247]][a, z]
Out[14]=	2 2 2 3 3 3 3 -12 -10 -2 5 z 5 z 11 z 10 z z 20 z 24 z z z -a - a - a + --- + --- + ----- + ----- + -- - ----- - ----- - -- + -- - 13 11 12 10 2 13 11 9 7 a a a a a a a a a 3 3 4 4 4 4 4 5 5 5 5 z z 25 z 19 z 3 z 2 z z 21 z 31 z 6 z 3 z > -- + -- - ----- - ----- + ---- - ---- + -- + ----- + ----- + ---- - ---- + 5 3 12 10 8 6 4 13 11 9 7 a a a a a a a a a a a 5 6 6 6 6 7 7 7 7 8 8 z 22 z 17 z 4 z z 8 z 14 z 5 z z 8 z 7 z > -- + ----- + ----- - ---- + -- - ---- - ----- - ---- + -- - ---- - ---- + 5 12 10 8 6 13 11 9 7 12 10 a a a a a a a a a a a 8 9 9 9 10 10 z z 2 z z z z > -- + --- + ---- + -- + --- + --- 8 13 11 9 12 10 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 247]], Vassiliev[3][Knot[11, Alternating, 247]]}
Out[15]=	{5, 15}
In[16]:=	Kh[Knot[11, Alternating, 247]][q, t]
Out[16]=	3 3 5 2 7 2 7 3 9 3 9 4 11 4 11 5 q + q + q t + q t + q t + q t + q t + q t + q t + q t + 13 5 13 6 15 6 15 7 17 7 17 8 19 8 21 9 > q t + q t + q t + q t + q t + q t + q t + q t + 21 10 25 11 > q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a247

K11a246 K11a248