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

K11n69 K11n71

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X5,14,6,15 X2837 X9,19,10,18 X11,21,12,20 X13,6,14,7 X15,22,16,1 X17,13,18,12 X19,11,20,10 X21,16,22,17

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 2t-2 - 2t-1 + 3 - 2t + 2t2 - t3

Conway Polynomial:

1 - 3z2 - 4z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{13, 4}

Jones Polynomial:

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

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

{81, ...}

A2 (sl(3)) Invariant:

q-6 + q-4 + 2q-2 + 1 - q4 - 2q6 - q8 - q10 + q14 + q16 + q18

HOMFLY-PT Polynomial:

3a-4 + 4a-4z2 + a-4z4 - 6a-2 - 11a-2z2 - 6a-2z4 - a-2z6 + 4 + 4z2 + z4

Kauffman Polynomial:

- a-7z + 4a-6z2 - 5a-6z4 + a-6z6 - 6a-5z + 15a-5z3 - 11a-5z5 + 2a-5z7 + 3a-4 - 9a-4z2 + 16a-4z4 - 11a-4z6 + 2a-4z8 - 6a-3z + 11a-3z3 - a-3z5 - 4a-3z7 + a-3z9 + 6a-2 - 27a-2z2 + 37a-2z4 - 19a-2z6 + 3a-2z8 - a-1z - 4a-1z3 + 10a-1z5 - 6a-1z7 + a-1z9 + 4 - 14z2 + 16z4 - 7z6 + z8

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

{-3, -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=4 is the signature of 1170. 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 j = 13                 1 j = 11                   j = 9             2 2   j = 7           1       j = 5         1 2 1     j = 3       2 2         j = 1       1 1         j = -1   1 2             j = -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, 70]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 70]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 14, 6, 15], X[2, 8, 3, 7], > X[9, 19, 10, 18], X[11, 21, 12, 20], X[13, 6, 14, 7], X[15, 22, 16, 1], > X[17, 13, 18, 12], X[19, 11, 20, 10], X[21, 16, 22, 17]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 70]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 7, 4, -2, -5, 10, -6, 9, -7, 3, -8, 11, -9, 5, -10, > 6, -11, 8]
In[5]:=	DTCode[Knot[11, NonAlternating, 70]]
Out[5]=	DTCode[4, 8, -14, 2, -18, -20, -6, -22, -12, -10, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 70]][t]
Out[6]=	-3 2 2 2 3 3 - t + -- - - - 2 t + 2 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 70]][z]
Out[7]=	2 4 6 1 - 3 z - 4 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 70]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 70]], KnotSignature[Knot[11, NonAlternating, 70]]}
Out[9]=	{13, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 70]][q]
Out[10]=	-2 1 2 3 4 5 6 2 + q - - - 2 q + 2 q - 2 q + q - q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[8, 1], Knot[11, NonAlternating, 70]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 70]][q]
Out[12]=	-6 -4 2 4 6 8 10 14 16 18 1 + q + q + -- - q - 2 q - q - q + q + q + q 2 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 70]][a, z]
Out[13]=	2 2 4 4 6 3 6 2 4 z 11 z 4 z 6 z z 4 + -- - -- + 4 z + ---- - ----- + z + -- - ---- - -- 4 2 4 2 4 2 2 a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 70]][a, z]
Out[14]=	2 2 2 3 3 6 z 6 z 6 z z 2 4 z 9 z 27 z 15 z 4 + -- + -- - -- - --- - --- - - - 14 z + ---- - ---- - ----- + ----- + 4 2 7 5 3 a 6 4 2 5 a a a a a a a a a 3 3 4 4 4 5 5 5 11 z 4 z 4 5 z 16 z 37 z 11 z z 10 z 6 > ----- - ---- + 16 z - ---- + ----- + ----- - ----- - -- + ----- - 7 z + 3 a 6 4 2 5 3 a a a a a a a 6 6 6 7 7 7 8 8 9 9 z 11 z 19 z 2 z 4 z 6 z 8 2 z 3 z z z > -- - ----- - ----- + ---- - ---- - ---- + z + ---- + ---- + -- + -- 6 4 2 5 3 a 4 2 3 a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 70]], Vassiliev[3][Knot[11, NonAlternating, 70]]}
Out[15]=	{-3, -3}
In[16]:=	Kh[Knot[11, NonAlternating, 70]][q, t]
Out[16]=	3 3 5 1 1 2 q 2 q 5 7 5 2 q + 2 q + q + ----- + ---- + ---- + - + ---- + 2 q t + q t + q t + 5 4 3 2 t t q t q t q t 9 2 9 3 13 4 > 2 q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n70

K11n69 K11n71