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

K11n163 K11n165

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X3,11,4,10 X14,5,15,6 X16,8,17,7 X9,21,10,20 X11,5,12,4 X13,19,14,18 X2,15,3,16 X22,18,1,17 X19,13,20,12 X21,9,22,8

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 5t-2 - 10t-1 + 13 - 10t + 5t2 - t3

Conway Polynomial:

1 + z2 - z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{818, 924, K11n85, ...}

Determinant and Signature:

{45, 4}

Jones Polynomial:

1 - 3q + 6q2 - 6q3 + 8q4 - 8q5 + 6q6 - 5q7 + 2q8

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

{...}

A2 (sl(3)) Invariant:

1 - q2 + q4 + q6 + 4q10 + 2q14 - 2q16 - 3q18 - q20 - 2q22 + 2q24 + q26

HOMFLY-PT Polynomial:

a-8 - 3a-6 + a-6z4 + 2a-4 - a-4z2 - 3a-4z4 - a-4z6 + a-2 + 2a-2z2 + a-2z4

Kauffman Polynomial:

2a-10z2 - 3a-9z + 4a-9z3 + a-9z5 + a-8 + 4a-8z4 - 2a-8z6 + a-8z8 - 7a-7z + 8a-7z3 - 2a-7z5 - a-7z7 + a-7z9 + 3a-6 - 3a-6z2 + 6a-6z4 - 10a-6z6 + 4a-6z8 - 4a-5z + 9a-5z3 - 12a-5z5 + 2a-5z7 + a-5z9 + 2a-4 + 2a-4z2 - a-4z4 - 7a-4z6 + 3a-4z8 + 5a-3z3 - 9a-3z5 + 3a-3z7 - a-2 + 3a-2z2 - 3a-2z4 + a-2z6

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

{1, 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=4 is the signature of 11164. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 17                 2 j = 15               3   j = 13             3 2   j = 11           5 3     j = 9         3 3       j = 7       3 5         j = 5     3 3           j = 3   1 4             j = 1   2               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, NonAlternating, 164]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 164]]
Out[3]=	PD[X[6, 2, 7, 1], X[3, 11, 4, 10], X[14, 5, 15, 6], X[16, 8, 17, 7], > X[9, 21, 10, 20], X[11, 5, 12, 4], X[13, 19, 14, 18], X[2, 15, 3, 16], > X[22, 18, 1, 17], X[19, 13, 20, 12], X[21, 9, 22, 8]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 164]]
Out[4]=	GaussCode[1, -8, -2, 6, 3, -1, 4, 11, -5, 2, -6, 10, -7, -3, 8, -4, 9, 7, -10, > 5, -11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 164]]
Out[5]=	DTCode[6, -10, 14, 16, -20, -4, -18, 2, 22, -12, -8]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 164]][t]
Out[6]=	-3 5 10 2 3 13 - t + -- - -- - 10 t + 5 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 164]][z]
Out[7]=	2 4 6 1 + z - z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 18], Knot[9, 24], Knot[11, NonAlternating, 85], > Knot[11, NonAlternating, 164]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 164]], KnotSignature[Knot[11, NonAlternating, 164]]}
Out[9]=	{45, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 164]][q]
Out[10]=	2 3 4 5 6 7 8 1 - 3 q + 6 q - 6 q + 8 q - 8 q + 6 q - 5 q + 2 q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 164]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 164]][q]
Out[12]=	2 4 6 10 14 16 18 20 22 24 26 1 - q + q + q + 4 q + 2 q - 2 q - 3 q - q - 2 q + 2 q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 164]][a, z]
Out[13]=	2 2 4 4 4 6 -8 3 2 -2 z 2 z z 3 z z z a - -- + -- + a - -- + ---- + -- - ---- + -- - -- 6 4 4 2 6 4 2 4 a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 164]][a, z]
Out[14]=	2 2 2 2 3 -8 3 2 -2 3 z 7 z 4 z 2 z 3 z 2 z 3 z 4 z a + -- + -- - a - --- - --- - --- + ---- - ---- + ---- + ---- + ---- + 6 4 9 7 5 10 6 4 2 9 a a a a a a a a a a 3 3 3 4 4 4 4 5 5 5 5 8 z 9 z 5 z 4 z 6 z z 3 z z 2 z 12 z 9 z > ---- + ---- + ---- + ---- + ---- - -- - ---- + -- - ---- - ----- - ---- - 7 5 3 8 6 4 2 9 7 5 3 a a a a a a a a a a a 6 6 6 6 7 7 7 8 8 8 9 9 2 z 10 z 7 z z z 2 z 3 z z 4 z 3 z z z > ---- - ----- - ---- + -- - -- + ---- + ---- + -- + ---- + ---- + -- + -- 8 6 4 2 7 5 3 8 6 4 7 5 a a a a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 164]], Vassiliev[3][Knot[11, NonAlternating, 164]]}
Out[15]=	{1, 0}
In[16]:=	Kh[Knot[11, NonAlternating, 164]][q, t]
Out[16]=	3 3 5 1 2 q q 5 7 7 2 9 2 9 3 4 q + 3 q + ---- + --- + -- + 3 q t + 3 q t + 5 q t + 3 q t + 3 q t + 2 t t q t 11 3 11 4 13 4 13 5 15 5 17 6 > 5 q t + 3 q t + 3 q t + 2 q t + 3 q t + 2 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n164

K11n163 K11n165