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

K11a102 K11a104

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 12 18 2 6 22 20 8 16

Alexander Polynomial:

4t-2 - 20t-1 + 33 - 20t + 4t2

Conway Polynomial:

1 - 4z2 + 4z4

Other knots with the same Alexander/Conway Polynomial:

{K11a201, ...}

Determinant and Signature:

{81, 0}

Jones Polynomial:

q-4 - 2q-3 + 5q-2 - 8q-1 + 11 - 13q + 12q2 - 11q3 + 9q4 - 5q5 + 3q6 - q7

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

{...}

A2 (sl(3)) Invariant:

q-14 + q-12 - q-10 + 2q-8 + q-6 - 3q-4 + 2q-2 - 1 - q2 - 2q6 + 2q8 - q10 + q12 + 3q14 - 2q16 + q18 + q20 - q22

HOMFLY-PT Polynomial:

- a-6z2 + a-4 + a-4z4 + a-2z2 + 2a-2z4 - 1 - 2z2 + z4 - 2a2z2 + a4

Kauffman Polynomial:

4a-7z3 - 4a-7z5 + a-7z7 - 7a-6z2 + 17a-6z4 - 13a-6z6 + 3a-6z8 + 2a-5z - 3a-5z3 + 8a-5z5 - 10a-5z7 + 3a-5z9 + a-4 - 16a-4z2 + 32a-4z4 - 25a-4z6 + 4a-4z8 + a-4z10 + 6a-3z - 13a-3z3 + 17a-3z5 - 18a-3z7 + 6a-3z9 - 7a-2z2 + 19a-2z4 - 20a-2z6 + 5a-2z8 + a-2z10 + 2a-1z + a-1z3 - 2a-1z5 - 3a-1z7 + 3a-1z9 - 1 + 4z2 + z4 - 5z6 + 4z8 - 2az + 5az3 - 5az5 + 4az7 - 2a2z4 + 3a2z6 - 2a3z3 + 2a3z5 + a4 - 2a4z2 + a4z4

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

{-4, 0}

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=0 is the signature of 11103. 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 = 15                       1 j = 13                     2   j = 11                   3 1   j = 9                 6 2     j = 7               5 3       j = 5             7 6         j = 3           6 5           j = 1         5 7             j = -1       4 7               j = -3     1 4                 j = -5   1 4                   j = -7   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, 103]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 103]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[14, 6, 15, 5], X[12, 8, 13, 7], > X[18, 9, 19, 10], X[2, 11, 3, 12], X[6, 14, 7, 13], X[22, 16, 1, 15], > X[20, 18, 21, 17], X[8, 19, 9, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 103]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -7, 4, -10, 5, -2, 6, -4, 7, -3, 8, -11, 9, -5, 10, > -9, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 103]]
Out[5]=	DTCode[4, 10, 14, 12, 18, 2, 6, 22, 20, 8, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 103]][t]
Out[6]=	4 20 2 33 + -- - -- - 20 t + 4 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 103]][z]
Out[7]=	2 4 1 - 4 z + 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 103], Knot[11, Alternating, 201]}
In[9]:=	{KnotDet[Knot[11, Alternating, 103]], KnotSignature[Knot[11, Alternating, 103]]}
Out[9]=	{81, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 103]][q]
Out[10]=	-4 2 5 8 2 3 4 5 6 7 11 + q - -- + -- - - - 13 q + 12 q - 11 q + 9 q - 5 q + 3 q - q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 103]}
In[12]:=	A2Invariant[Knot[11, Alternating, 103]][q]
Out[12]=	-14 -12 -10 2 -6 3 2 2 6 8 10 12 -1 + q + q - q + -- + q - -- + -- - q - 2 q + 2 q - q + q + 8 4 2 q q q 14 16 18 20 22 > 3 q - 2 q + q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 103]][a, z]
Out[13]=	2 2 4 4 -4 4 2 z z 2 2 4 z 2 z -1 + a + a - 2 z - -- + -- - 2 a z + z + -- + ---- 6 2 4 2 a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 103]][a, z]
Out[14]=	2 2 2 -4 4 2 z 6 z 2 z 2 7 z 16 z 7 z -1 + a + a + --- + --- + --- - 2 a z + 4 z - ---- - ----- - ---- - 5 3 a 6 4 2 a a a a a 3 3 3 3 4 4 2 4 z 3 z 13 z z 3 3 3 4 17 z > 2 a z + ---- - ---- - ----- + -- + 5 a z - 2 a z + z + ----- + 7 5 3 a 6 a a a a 4 4 5 5 5 5 32 z 19 z 2 4 4 4 4 z 8 z 17 z 2 z 5 > ----- + ----- - 2 a z + a z - ---- + ---- + ----- - ---- - 5 a z + 4 2 7 5 3 a a a a a a 6 6 6 7 7 7 3 5 6 13 z 25 z 20 z 2 6 z 10 z 18 z > 2 a z - 5 z - ----- - ----- - ----- + 3 a z + -- - ----- - ----- - 6 4 2 7 5 3 a a a a a a 7 8 8 8 9 9 9 10 10 3 z 7 8 3 z 4 z 5 z 3 z 6 z 3 z z z > ---- + 4 a z + 4 z + ---- + ---- + ---- + ---- + ---- + ---- + --- + --- a 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 103]], Vassiliev[3][Knot[11, Alternating, 103]]}
Out[15]=	{-4, 0}
In[16]:=	Kh[Knot[11, Alternating, 103]][q, t]
Out[16]=	7 1 1 1 4 1 4 4 3 - + 5 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 7 q t + 6 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 > 5 q t + 7 q t + 6 q t + 5 q t + 3 q t + 6 q t + 2 q t + 11 5 11 6 13 6 15 7 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a103

K11a102 K11a104