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

K11n68 K11n70

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X5,15,6,14 X2837 X9,19,10,18 X11,17,12,16 X13,20,14,21 X15,7,16,6 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-3 + 7t-2 - 9t-1 + 9 - 9t + 7t2 - 2t3

Conway Polynomial:

1 + z2 - 5z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{1021, ...}

Determinant and Signature:

{45, 4}

Jones Polynomial:

1 - 2q + 5q2 - 6q3 + 7q4 - 8q5 + 7q6 - 5q7 + 3q8 - q9

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

{...}

A2 (sl(3)) Invariant:

1 + q4 + 2q6 + 2q10 - 2q12 - q16 - 2q18 + q20 - q22 + 2q24 + q26 - q32

HOMFLY-PT Polynomial:

- a-10 + 3a-8 + 4a-8z2 + a-8z4 - 3a-6 - 5a-6z2 - 4a-6z4 - a-6z6 - a-4z2 - 3a-4z4 - a-4z6 + 2a-2 + 3a-2z2 + a-2z4

Kauffman Polynomial:

- a-11z + a-11z3 + a-10 - 3a-10z2 + 3a-10z4 - a-9z + 4a-9z3 - a-9z5 + a-9z7 + 3a-8 - 12a-8z2 + 16a-8z4 - 7a-8z6 + 2a-8z8 + a-7z - a-7z3 + a-7z5 - a-7z7 + a-7z9 + 3a-6 - 12a-6z2 + 15a-6z4 - 12a-6z6 + 4a-6z8 + 2a-5z - a-5z3 - 4a-5z5 + a-5z9 + 2a-4z2 - 2a-4z4 - 4a-4z6 + 2a-4z8 + a-3z + 3a-3z3 - 6a-3z5 + 2a-3z7 - 2a-2 + 5a-2z2 - 4a-2z4 + a-2z6

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

{1, 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=4 is the signature of 1169. 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 = 19                   1 j = 17                 2   j = 15               3 1   j = 13             4 2     j = 11           4 3       j = 9         3 4         j = 7       3 4           j = 5     2 3             j = 3   1 4               j = 1   1                 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, 69]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 69]]
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, 19, 10, 18], X[11, 17, 12, 16], X[13, 20, 14, 21], X[15, 7, 16, 6], > X[17, 11, 18, 10], X[19, 1, 20, 22], X[21, 12, 22, 13]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 69]]
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, 69]]
Out[5]=	DTCode[4, 8, -14, 2, -18, -16, -20, -6, -10, -22, -12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 69]][t]
Out[6]=	2 7 9 2 3 9 - -- + -- - - - 9 t + 7 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 69]][z]
Out[7]=	2 4 6 1 + z - 5 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 21], Knot[11, NonAlternating, 69]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 69]], KnotSignature[Knot[11, NonAlternating, 69]]}
Out[9]=	{45, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 69]][q]
Out[10]=	2 3 4 5 6 7 8 9 1 - 2 q + 5 q - 6 q + 7 q - 8 q + 7 q - 5 q + 3 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 69]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 69]][q]
Out[12]=	4 6 10 12 16 18 20 22 24 26 32 1 + q + 2 q + 2 q - 2 q - q - 2 q + q - q + 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 69]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 6 6 -10 3 3 2 4 z 5 z z 3 z z 4 z 3 z z z z -a + -- - -- + -- + ---- - ---- - -- + ---- + -- - ---- - ---- + -- - -- - -- 8 6 2 8 6 4 2 8 6 4 2 6 4 a a a a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 69]][a, z]
Out[14]=	2 2 2 2 -10 3 3 2 z z z 2 z z 3 z 12 z 12 z 2 z a + -- + -- - -- - --- - -- + -- + --- + -- - ---- - ----- - ----- + ---- + 8 6 2 11 9 7 5 3 10 8 6 4 a a a a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 4 5 z z 4 z z z 3 z 3 z 16 z 15 z 2 z 4 z > ---- + --- + ---- - -- - -- + ---- + ---- + ----- + ----- - ---- - ---- - 2 11 9 7 5 3 10 8 6 4 2 a a a a a a a a a a a 5 5 5 5 6 6 6 6 7 7 7 8 z z 4 z 6 z 7 z 12 z 4 z z z z 2 z 2 z > -- + -- - ---- - ---- - ---- - ----- - ---- + -- + -- - -- + ---- + ---- + 9 7 5 3 8 6 4 2 9 7 3 8 a a a a a a a a a a a a 8 8 9 9 4 z 2 z z z > ---- + ---- + -- + -- 6 4 7 5 a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 69]], Vassiliev[3][Knot[11, NonAlternating, 69]]}
Out[15]=	{1, 2}
In[16]:=	Kh[Knot[11, NonAlternating, 69]][q, t]
Out[16]=	3 3 5 1 q q 5 7 7 2 9 2 9 3 4 q + 2 q + ---- + - + -- + 3 q t + 3 q t + 4 q t + 3 q t + 4 q t + 2 t t q t 11 3 11 4 13 4 13 5 15 5 15 6 17 6 > 4 q t + 3 q t + 4 q t + 2 q t + 3 q t + q t + 2 q t + 19 7 > q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n69

K11n68 K11n70