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

K11n67 K11n69

Knotscape

This page is passe. Go here instead!    The Knot K11n68 Visit K11n68's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X4251 X8493 X5,15,6,14 X2837 X18,9,19,10 X16,11,17,12 X20,14,21,13 X15,7,16,6 X10,17,11,18 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:

- 4t-2 + 16t-1 - 23 + 16t - 4t2

Conway Polynomial:

1 - 4z4

Other knots with the same Alexander/Conway Polynomial:

{1067, 1074, ...}

Determinant and Signature:

{63, 2}

Jones Polynomial:

2q-1 - 4 + 7q - 10q2 + 11q3 - 10q4 + 9q5 - 6q6 + 3q7 - q8

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

{...}

A2 (sl(3)) Invariant:

2q-4 + 3q2 - 3q4 - q8 - q10 + 2q12 - q14 + 3q16 - q20 + 2q22 - q24 - q26

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-9z - 2a-9z3 + a-9z5 - a-8 + 3a-8z2 - 6a-8z4 + 3a-8z6 - 6a-7z5 + 4a-7z7 - 2a-6 + 6a-6z2 - 7a-6z4 - a-6z6 + 3a-6z8 - 2a-5z + 9a-5z3 - 12a-5z5 + 5a-5z7 + a-5z9 - a-4z2 + 6a-4z4 - 8a-4z6 + 5a-4z8 + 4a-3z3 - 4a-3z5 + 2a-3z7 + a-3z9 + 2a-2 - 10a-2z2 + 10a-2z4 - 4a-2z6 + 2a-2z8 + a-1z - 3a-1z3 + a-1z5 + a-1z7 + 2 - 6z2 + 3z4

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

{0, 2}

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 1168. 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 r = 7 j = 17                   1 j = 15                 2   j = 13               4 1   j = 11             5 2     j = 9           5 4       j = 7         6 5         j = 5       4 5           j = 3     3 6             j = 1   2 5               j = -1   2                 j = -3 2

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, 68]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 68]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 15, 6, 14], X[2, 8, 3, 7], > X[18, 9, 19, 10], X[16, 11, 17, 12], X[20, 14, 21, 13], X[15, 7, 16, 6], > X[10, 17, 11, 18], X[22, 19, 1, 20], X[12, 22, 13, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 68]]
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, 68]]
Out[5]=	DTCode[4, 8, -14, 2, 18, 16, 20, -6, 10, 22, 12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 68]][t]
Out[6]=	4 16 2 -23 - -- + -- + 16 t - 4 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 68]][z]
Out[7]=	4 1 - 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 67], Knot[10, 74], Knot[11, NonAlternating, 68]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 68]], KnotSignature[Knot[11, NonAlternating, 68]]}
Out[9]=	{63, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 68]][q]
Out[10]=	2 2 3 4 5 6 7 8 -4 + - + 7 q - 10 q + 11 q - 10 q + 9 q - 6 q + 3 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 68]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 68]][q]
Out[12]=	2 2 4 8 10 12 14 16 20 22 24 26 -- + 3 q - 3 q - q - q + 2 q - q + 3 q - q + 2 q - q - q 4 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 68]][a, z]
Out[13]=	2 2 2 4 4 -8 2 2 2 3 z 2 z 3 z 2 z 2 z 2 - a + -- - -- + 2 z + ---- - ---- - ---- - ---- - ---- 6 2 6 4 2 4 2 a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 68]][a, z]
Out[14]=	2 2 2 2 3 -8 2 2 z 2 z z 2 3 z 6 z z 10 z 2 z 2 - a - -- + -- + -- - --- + - - 6 z + ---- + ---- - -- - ----- - ---- + 6 2 9 5 a 8 6 4 2 9 a a a a a a a a a 3 3 3 4 4 4 4 5 5 9 z 4 z 3 z 4 6 z 7 z 6 z 10 z z 6 z > ---- + ---- - ---- + 3 z - ---- - ---- + ---- + ----- + -- - ---- - 5 3 a 8 6 4 2 9 7 a a a a a a a a 5 5 5 6 6 6 6 7 7 7 7 12 z 4 z z 3 z z 8 z 4 z 4 z 5 z 2 z z > ----- - ---- + -- + ---- - -- - ---- - ---- + ---- + ---- + ---- + -- + 5 3 a 8 6 4 2 7 5 3 a a a a a a a a a a 8 8 8 9 9 3 z 5 z 2 z z z > ---- + ---- + ---- + -- + -- 6 4 2 5 3 a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 68]], Vassiliev[3][Knot[11, NonAlternating, 68]]}
Out[15]=	{0, 2}
In[16]:=	Kh[Knot[11, NonAlternating, 68]][q, t]
Out[16]=	3 2 2 2 q 3 5 5 2 7 2 5 q + 3 q + ----- + --- + --- + 6 q t + 4 q t + 5 q t + 6 q t + 3 2 q t t q t 7 3 9 3 9 4 11 4 11 5 13 5 13 6 > 5 q t + 5 q t + 4 q t + 5 q t + 2 q t + 4 q t + q t + 15 6 17 7 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n68

K11n67 K11n69