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

K11n65 K11n67

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X14,5,15,6 X2837 X9,19,10,18 X11,17,12,16 X20,14,21,13 X6,15,7,16 X17,11,18,10 X22,19,1,20 X12,22,13,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 14 2 -18 -16 20 6 -10 22 12

Alexander Polynomial:

t-3 - 7t-2 + 18t-1 - 23 + 18t - 7t2 + t3

Conway Polynomial:

1 - z2 - z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{940, 1059, ...}

Determinant and Signature:

{75, 2}

Jones Polynomial:

- q-2 + 4q-1 - 7 + 11q - 12q2 + 13q3 - 12q4 + 8q5 - 5q6 + 2q7

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

{K11n150, ...}

A2 (sl(3)) Invariant:

- q-6 + 2q-4 - q-2 + 4q2 - q4 + 4q6 - q8 - q10 - q12 - 4q14 + 2q16 - q18 + 2q22

HOMFLY-PT Polynomial:

2a-6 + 2a-6z2 - 5a-4 - 7a-4z2 - 3a-4z4 + 4a-2 + 5a-2z2 + 3a-2z4 + a-2z6 - z2 - z4

Kauffman Polynomial:

- 4a-8z2 + 3a-8z4 + 3a-7z - 6a-7z3 + 3a-7z5 + a-7z7 - 2a-6 + a-6z2 + 2a-6z8 + 4a-5z - 7a-5z5 + 5a-5z7 + a-5z9 - 5a-4 + 14a-4z2 - 11a-4z4 - 3a-4z6 + 6a-4z8 + 10a-3z3 - 21a-3z5 + 10a-3z7 + a-3z9 - 4a-2 + 12a-2z2 - 15a-2z4 + a-2z6 + 4a-2z8 - a-1z + 3a-1z3 - 10a-1z5 + 6a-1z7 + 3z2 - 7z4 + 4z6 - az3 + az5

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

{-1, -3}

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 1166. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 15                   2 j = 13                 3   j = 11               5 2   j = 9             7 3     j = 7           6 5       j = 5         6 7         j = 3       5 6           j = 1     3 7             j = -1   1 4               j = -3   3                 j = -5 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, NonAlternating, 66]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 66]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[14, 5, 15, 6], X[2, 8, 3, 7], > X[9, 19, 10, 18], X[11, 17, 12, 16], X[20, 14, 21, 13], X[6, 15, 7, 16], > X[17, 11, 18, 10], X[22, 19, 1, 20], X[12, 22, 13, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 66]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -8, 4, -2, -5, 9, -6, -11, 7, -3, 8, 6, -9, 5, 10, > -7, 11, -10]
In[5]:=	DTCode[Knot[11, NonAlternating, 66]]
Out[5]=	DTCode[4, 8, 14, 2, -18, -16, 20, 6, -10, 22, 12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 66]][t]
Out[6]=	-3 7 18 2 3 -23 + t - -- + -- + 18 t - 7 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 66]][z]
Out[7]=	2 4 6 1 - z - z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 40], Knot[10, 59], Knot[11, NonAlternating, 66]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 66]], KnotSignature[Knot[11, NonAlternating, 66]]}
Out[9]=	{75, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 66]][q]
Out[10]=	-2 4 2 3 4 5 6 7 -7 - q + - + 11 q - 12 q + 13 q - 12 q + 8 q - 5 q + 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 66], Knot[11, NonAlternating, 150]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 66]][q]
Out[12]=	-6 2 -2 2 4 6 8 10 12 14 16 18 -q + -- - q + 4 q - q + 4 q - q - q - q - 4 q + 2 q - q + 4 q 22 > 2 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 66]][a, z]
Out[13]=	2 2 2 4 4 6 2 5 4 2 2 z 7 z 5 z 4 3 z 3 z z -- - -- + -- - z + ---- - ---- + ---- - z - ---- + ---- + -- 6 4 2 6 4 2 4 2 2 a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 66]][a, z]
Out[14]=	2 2 2 2 3 -2 5 4 3 z 4 z z 2 4 z z 14 z 12 z 6 z -- - -- - -- + --- + --- - - + 3 z - ---- + -- + ----- + ----- - ---- + 6 4 2 7 5 a 8 6 4 2 7 a a a a a a a a a a 3 3 4 4 4 5 5 5 10 z 3 z 3 4 3 z 11 z 15 z 3 z 7 z 21 z > ----- + ---- - a z - 7 z + ---- - ----- - ----- + ---- - ---- - ----- - 3 a 8 4 2 7 5 3 a a a a a a a 5 6 6 7 7 7 7 8 8 10 z 5 6 3 z z z 5 z 10 z 6 z 2 z 6 z > ----- + a z + 4 z - ---- + -- + -- + ---- + ----- + ---- + ---- + ---- + a 4 2 7 5 3 a 6 4 a a a a a a a 8 9 9 4 z z z > ---- + -- + -- 2 5 3 a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 66]], Vassiliev[3][Knot[11, NonAlternating, 66]]}
Out[15]=	{-1, -3}
In[16]:=	Kh[Knot[11, NonAlternating, 66]][q, t]
Out[16]=	3 1 3 1 4 3 q 3 5 5 2 7 q + 5 q + ----- + ----- + ---- + --- + --- + 6 q t + 6 q t + 7 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 13 5 > 6 q t + 5 q t + 7 q t + 3 q t + 5 q t + 2 q t + 3 q t + 15 6 > 2 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n66

K11n65 K11n67