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

K11n66 K11n68

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X14,5,15,6 X2837 X9,19,10,18 X11,17,12,16 X13,20,14,21 X6,15,7,16 X17,11,18,10 X19,1,20,22 X21,12,22,13

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:

- 2t-1 + 5 - 2t

Conway Polynomial:

1 - 2z2

Other knots with the same Alexander/Conway Polynomial:

{61, 946, K11n97, K11n139, ...}

Determinant and Signature:

{9, 0}

Jones Polynomial:

q-3 - q-2 + q-1 - q + q2 - 2q3 + 3q4 - 2q5 + 2q6 - q7

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

{...}

A2 (sl(3)) Invariant:

q-10 + q-8 + q-6 - 2q2 - q4 - q6 + 2q10 + q12 + 2q14 - q22

HOMFLY-PT Polynomial:

- a-6 - a-6z2 + 3a-4 + 3a-4z2 + a-4z4 - a-2 - a-2z2 - 2 - 4z2 - z4 + 2a2 + a2z2

Kauffman Polynomial:

- a-7z + 6a-7z3 - 5a-7z5 + a-7z7 + a-6 - 9a-6z2 + 17a-6z4 - 11a-6z6 + 2a-6z8 + 3a-5z3 + a-5z5 - 4a-5z7 + a-5z9 + 3a-4 - 20a-4z2 + 31a-4z4 - 18a-4z6 + 3a-4z8 + 4a-3z - 9a-3z3 + 10a-3z5 - 6a-3z7 + a-3z9 + a-2 - 8a-2z2 + 12a-2z4 - 7a-2z6 + a-2z8 + 2a-1z - a-1z3 - a-1z5 - 2 + 9z2 - 7z4 + z6 - az + 5az3 - 5az5 + az7 - 2a2 + 6a2z2 - 5a2z4 + a2z6

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

{-2, 1}

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 1167. 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                     1   j = 11                   1 1   j = 9                 2 1     j = 7             1 1 1       j = 5             1 2         j = 3         1 2 1           j = 1       1 1 1             j = -1       1 2               j = -3   1 1                   j = -5                         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, NonAlternating, 67]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 67]]
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[13, 20, 14, 21], X[6, 15, 7, 16], > X[17, 11, 18, 10], X[19, 1, 20, 22], X[21, 12, 22, 13]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 67]]
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, 67]]
Out[5]=	DTCode[4, 8, 14, 2, -18, -16, -20, 6, -10, -22, -12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 67]][t]
Out[6]=	2 5 - - - 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 67]][z]
Out[7]=	2 1 - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[6, 1], Knot[9, 46], Knot[11, NonAlternating, 67], > Knot[11, NonAlternating, 97], Knot[11, NonAlternating, 139]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 67]], KnotSignature[Knot[11, NonAlternating, 67]]}
Out[9]=	{9, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 67]][q]
Out[10]=	-3 -2 1 2 3 4 5 6 7 q - q + - - q + q - 2 q + 3 q - 2 q + 2 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 67]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 67]][q]
Out[12]=	-10 -8 -6 2 4 6 10 12 14 22 q + q + q - 2 q - q - q + 2 q + q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 67]][a, z]
Out[13]=	2 2 2 4 -6 3 -2 2 2 z 3 z z 2 2 4 z -2 - a + -- - a + 2 a - 4 z - -- + ---- - -- + a z - z + -- 4 6 4 2 4 a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 67]][a, z]
Out[14]=	2 2 -6 3 -2 2 z 4 z 2 z 2 9 z 20 z -2 + a + -- + a - 2 a - -- + --- + --- - a z + 9 z - ---- - ----- - 4 7 3 a 6 4 a a a a a 2 3 3 3 3 4 4 8 z 2 2 6 z 3 z 9 z z 3 4 17 z 31 z > ---- + 6 a z + ---- + ---- - ---- - -- + 5 a z - 7 z + ----- + ----- + 2 7 5 3 a 6 4 a a a a a a 4 5 5 5 5 6 6 12 z 2 4 5 z z 10 z z 5 6 11 z 18 z > ----- - 5 a z - ---- + -- + ----- - -- - 5 a z + z - ----- - ----- - 2 7 5 3 a 6 4 a a a a a a 6 7 7 7 8 8 8 9 9 7 z 2 6 z 4 z 6 z 7 2 z 3 z z z z > ---- + a z + -- - ---- - ---- + a z + ---- + ---- + -- + -- + -- 2 7 5 3 6 4 2 5 3 a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 67]], Vassiliev[3][Knot[11, NonAlternating, 67]]}
Out[15]=	{-2, 1}
In[16]:=	Kh[Knot[11, NonAlternating, 67]][q, t]
Out[16]=	2 3 1 1 1 1 q 3 3 2 5 2 - + q + q + ----- + ----- + ----- + --- + - + q t + 2 q t + q t + q t + q 7 4 3 3 3 2 q t t q t q t q t 7 2 5 3 7 3 7 4 9 4 9 5 11 5 11 6 > q t + 2 q t + q t + q t + 2 q t + q t + q t + q t + 13 6 15 7 > q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n67

K11n66 K11n68