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

K11n63 K11n65

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

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

Conway Polynomial:

1 - z2 - 7z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{21, 4}

Jones Polynomial:

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

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

{K11n28, ...}

A2 (sl(3)) Invariant:

q-6 + q-4 + q-2 + 1 - q4 - 2q6 - q8 + 2q14 + q16 + q18 - q26

HOMFLY-PT Polynomial:

- a-8 + 2a-6 + 4a-6z2 + a-6z4 + a-4 - 2a-4z2 - 4a-4z4 - a-4z6 - 4a-2 - 7a-2z2 - 5a-2z4 - a-2z6 + 3 + 4z2 + z4

Kauffman Polynomial:

a-9z - a-8 + 2a-8z2 + a-7z - 2a-7z3 + a-7z5 - 2a-6 + 6a-6z2 - 7a-6z4 + 2a-6z6 + 4a-5z3 - 7a-5z5 + 2a-5z7 + a-4 - 6a-4z2 + 11a-4z4 - 9a-4z6 + 2a-4z8 + 2a-3z - a-3z3 + 3a-3z5 - 4a-3z7 + a-3z9 + 4a-2 - 23a-2z2 + 34a-2z4 - 18a-2z6 + 3a-2z8 + 2a-1z - 7a-1z3 + 11a-1z5 - 6a-1z7 + a-1z9 + 3 - 13z2 + 16z4 - 7z6 + z8

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

{-1, 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=4 is the signature of 1164. 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 j = 15                   1 j = 13                 1   j = 11               1 1   j = 9             2 1     j = 7           1 1       j = 5         2 2         j = 3       2 2           j = 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, 64]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 64]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 15, 6, 14], X[2, 8, 3, 7], > X[9, 17, 10, 16], X[11, 20, 12, 21], X[13, 18, 14, 19], X[15, 7, 16, 6], > X[17, 1, 18, 22], X[19, 12, 20, 13], X[21, 10, 22, 11]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 64]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 8, 4, -2, -5, 11, -6, 10, -7, 3, -8, 5, -9, 7, -10, > 6, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 64]]
Out[5]=	DTCode[4, 8, -14, 2, -16, -20, -18, -6, -22, -12, -10]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 64]][t]
Out[6]=	2 5 3 2 3 1 - -- + -- - - - 3 t + 5 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 64]][z]
Out[7]=	2 4 6 1 - z - 7 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 64]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 64]], KnotSignature[Knot[11, NonAlternating, 64]]}
Out[9]=	{21, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 64]][q]
Out[10]=	-2 1 2 3 4 5 6 7 2 + q - - - 3 q + 3 q - 3 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, 28], Knot[11, NonAlternating, 64]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 64]][q]
Out[12]=	-6 -4 -2 4 6 8 14 16 18 26 1 + q + q + q - q - 2 q - q + 2 q + q + q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 64]][a, z]
Out[13]=	2 2 2 4 4 4 -8 2 -4 4 2 4 z 2 z 7 z 4 z 4 z 5 z 3 - a + -- + a - -- + 4 z + ---- - ---- - ---- + z + -- - ---- - ---- - 6 2 6 4 2 6 4 2 a a a a a a a a 6 6 z z > -- - -- 4 2 a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 64]][a, z]
Out[14]=	2 2 2 -8 2 -4 4 z z 2 z 2 z 2 2 z 6 z 6 z 3 - a - -- + a + -- + -- + -- + --- + --- - 13 z + ---- + ---- - ---- - 6 2 9 7 3 a 8 6 4 a a a a a a a a 2 3 3 3 3 4 4 4 5 23 z 2 z 4 z z 7 z 4 7 z 11 z 34 z z > ----- - ---- + ---- - -- - ---- + 16 z - ---- + ----- + ----- + -- - 2 7 5 3 a 6 4 2 7 a a a a a a a a 5 5 5 6 6 6 7 7 7 7 z 3 z 11 z 6 2 z 9 z 18 z 2 z 4 z 6 z > ---- + ---- + ----- - 7 z + ---- - ---- - ----- + ---- - ---- - ---- + 5 3 a 6 4 2 5 3 a a a a a a a a 8 8 9 9 8 2 z 3 z z z > z + ---- + ---- + -- + -- 4 2 3 a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 64]], Vassiliev[3][Knot[11, NonAlternating, 64]]}
Out[15]=	{-1, 1}
In[16]:=	Kh[Knot[11, NonAlternating, 64]][q, t]
Out[16]=	3 3 5 1 1 2 q 2 q 5 7 7 2 2 q + 2 q + ----- + ---- + ---- + - + ---- + 2 q t + q t + q t + 5 4 3 2 t t q t q t q t 9 2 9 3 11 3 11 4 13 4 15 5 > 2 q t + q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n64

K11n63 K11n65