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

K11a201 K11a203

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 16 18 14 20 2 8 6 22 10

Alexander Polynomial:

2t-3 - 12t-2 + 26t-1 - 31 + 26t - 12t2 + 2t3

Conway Polynomial:

1 - 4z2 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a137, ...}

Determinant and Signature:

{111, 2}

Jones Polynomial:

q-3 - 2q-2 + 6q-1 - 10 + 14q - 18q2 + 18q3 - 16q4 + 13q5 - 8q6 + 4q7 - q8

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

{...}

A2 (sl(3)) Invariant:

q-10 + q-8 + 3q-4 - q-2 - 1 + 2q2 - 5q4 + q6 - 2q8 + 3q12 - 2q14 + 4q16 - q18 - q20 + 2q22 - q24

HOMFLY-PT Polynomial:

- a-6z2 - a-6z4 + 2a-4 + 2a-4z2 + 2a-4z4 + a-4z6 - 2a-2 - 2a-2z2 + a-2z4 + a-2z6 - 1 - 4z2 - 2z4 + 2a2 + a2z2

Kauffman Polynomial:

- a-9z3 + a-9z5 + 2a-8z2 - 6a-8z4 + 4a-8z6 - a-7z + 4a-7z3 - 11a-7z5 + 7a-7z7 + 2a-6z2 - a-6z4 - 8a-6z6 + 7a-6z8 - 2a-5z + 11a-5z3 - 15a-5z5 + 3a-5z7 + 4a-5z9 + 2a-4 - 13a-4z2 + 27a-4z4 - 27a-4z6 + 11a-4z8 + a-4z10 + 4a-3z - 6a-3z3 + 8a-3z5 - 11a-3z7 + 7a-3z9 + 2a-2 - 17a-2z2 + 28a-2z4 - 22a-2z6 + 7a-2z8 + a-2z10 + 6a-1z - 10a-1z3 + 6a-1z5 - 5a-1z7 + 3a-1z9 - 1 + z2 + 2z4 - 6z6 + 3z8 + az + 2az3 - 5az5 + 2az7 - 2a2 + 5a2z2 - 4a2z4 + a2z6

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

{-4, -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 11202. 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                   5 1   j = 11                 8 3     j = 9               8 5       j = 7             10 8         j = 5           8 8           j = 3         6 10             j = 1       5 9               j = -1     1 5                 j = -3   1 5                   j = -5   1                     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, 202]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 202]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[16, 6, 17, 5], X[18, 8, 19, 7], > X[14, 10, 15, 9], X[20, 11, 21, 12], X[2, 13, 3, 14], X[8, 16, 9, 15], > X[6, 18, 7, 17], X[22, 20, 1, 19], X[10, 21, 11, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 202]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -9, 4, -8, 5, -11, 6, -2, 7, -5, 8, -3, 9, -4, 10, > -6, 11, -10]
In[5]:=	DTCode[Knot[11, Alternating, 202]]
Out[5]=	DTCode[4, 12, 16, 18, 14, 20, 2, 8, 6, 22, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 202]][t]
Out[6]=	2 12 26 2 3 -31 + -- - -- + -- + 26 t - 12 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 202]][z]
Out[7]=	2 6 1 - 4 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 137], Knot[11, Alternating, 202]}
In[9]:=	{KnotDet[Knot[11, Alternating, 202]], KnotSignature[Knot[11, Alternating, 202]]}
Out[9]=	{111, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 202]][q]
Out[10]=	-3 2 6 2 3 4 5 6 7 8 -10 + q - -- + - + 14 q - 18 q + 18 q - 16 q + 13 q - 8 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, 202]}
In[12]:=	A2Invariant[Knot[11, Alternating, 202]][q]
Out[12]=	-10 -8 3 -2 2 4 6 8 12 14 16 -1 + q + q + -- - q + 2 q - 5 q + q - 2 q + 3 q - 2 q + 4 q - 4 q 18 20 22 24 > q - q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 202]][a, z]
Out[13]=	2 2 2 4 4 4 2 2 2 2 z 2 z 2 z 2 2 4 z 2 z z -1 + -- - -- + 2 a - 4 z - -- + ---- - ---- + a z - 2 z - -- + ---- + -- + 4 2 6 4 2 6 4 2 a a a a a a a a 6 6 z z > -- + -- 4 2 a a
In[14]:=	Kauffman[Knot[11, Alternating, 202]][a, z]
Out[14]=	2 2 2 2 2 2 z 2 z 4 z 6 z 2 2 z 2 z 13 z -1 + -- + -- - 2 a - -- - --- + --- + --- + a z + z + ---- + ---- - ----- - 4 2 7 5 3 a 8 6 4 a a a a a a a a 2 3 3 3 3 3 4 17 z 2 2 z 4 z 11 z 6 z 10 z 3 4 6 z > ----- + 5 a z - -- + ---- + ----- - ---- - ----- + 2 a z + 2 z - ---- - 2 9 7 5 3 a 8 a a a a a a 4 4 4 5 5 5 5 5 z 27 z 28 z 2 4 z 11 z 15 z 8 z 6 z 5 > -- + ----- + ----- - 4 a z + -- - ----- - ----- + ---- + ---- - 5 a z - 6 4 2 9 7 5 3 a a a a a a a a 6 6 6 6 7 7 7 7 6 4 z 8 z 27 z 22 z 2 6 7 z 3 z 11 z 5 z > 6 z + ---- - ---- - ----- - ----- + a z + ---- + ---- - ----- - ---- + 8 6 4 2 7 5 3 a a a a a a a a 8 8 8 9 9 9 10 10 7 8 7 z 11 z 7 z 4 z 7 z 3 z z z > 2 a z + 3 z + ---- + ----- + ---- + ---- + ---- + ---- + --- + --- 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 202]], Vassiliev[3][Knot[11, Alternating, 202]]}
Out[15]=	{-4, -2}
In[16]:=	Kh[Knot[11, Alternating, 202]][q, t]
Out[16]=	3 1 1 1 5 1 5 5 q 3 9 q + 6 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 10 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 > 8 q t + 8 q t + 10 q t + 8 q t + 8 q t + 5 q t + 8 q t + 11 5 13 5 13 6 15 6 17 7 > 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a202

K11a201 K11a203