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

K11a67 K11a69

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X16,5,17,6 X14,8,15,7 X2,9,3,10 X18,12,19,11 X20,14,21,13 X22,15,1,16 X10,18,11,17 X12,20,13,19 X6,21,7,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 16 14 2 18 20 22 10 12 6

Alexander Polynomial:

- t-4 + 6t-3 - 14t-2 + 20t-1 - 21 + 20t - 14t2 + 6t3 - t4

Conway Polynomial:

1 + 2z2 + 2z4 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{103, 2}

Jones Polynomial:

- q-4 + 3q-3 - 6q-2 + 10q-1 - 13 + 16q - 16q2 + 15q3 - 11q4 + 7q5 - 4q6 + q7

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

{K11a111, ...}

A2 (sl(3)) Invariant:

- q-12 + q-8 - q-6 + 2q-4 - 2q-2 + 1 + 2q2 - q4 + 5q6 - 2q8 + 2q10 - q12 - 2q14 + q16 - 2q18 + q20

HOMFLY-PT Polynomial:

- a-4 + a-4z2 + 3a-4z4 + a-4z6 + a-2 - 4a-2z2 - 8a-2z4 - 5a-2z6 - a-2z8 + 2 + 8z2 + 8z4 + 2z6 - a2 - 3a2z2 - a2z4

Kauffman Polynomial:

a-8z4 - 3a-7z3 + 4a-7z5 + a-6z2 - 7a-6z4 + 7a-6z6 + 2a-5z - 3a-5z3 - 7a-5z5 + 8a-5z7 - a-4 + 2a-4z2 + 2a-4z4 - 11a-4z6 + 8a-4z8 + 3a-3z - 4a-3z3 + 5a-3z5 - 10a-3z7 + 6a-3z9 - a-2 - 7a-2z2 + 30a-2z4 - 30a-2z6 + 6a-2z8 + 2a-2z10 + a-1z - 10a-1z3 + 32a-1z5 - 33a-1z7 + 10a-1z9 + 2 - 15z2 + 34z4 - 24z6 + z8 + 2z10 - az - 2az3 + 12az5 - 14az7 + 4az9 + a2 - 7a2z2 + 14a2z4 - 12a2z6 + 3a2z8 - a3z + 4a3z3 - 4a3z5 + a3z7

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

{2, 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=2 is the signature of 1168. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 15                       1 j = 13                     3   j = 11                   4 1   j = 9                 7 3     j = 7               8 4       j = 5             8 7         j = 3           8 8           j = 1         6 9             j = -1       4 7               j = -3     2 6                 j = -5   1 4                   j = -7   2                     j = -9 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, Alternating, 68]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 68]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[16, 5, 17, 6], X[14, 8, 15, 7], > X[2, 9, 3, 10], X[18, 12, 19, 11], X[20, 14, 21, 13], X[22, 15, 1, 16], > X[10, 18, 11, 17], X[12, 20, 13, 19], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 68]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -11, 4, -2, 5, -9, 6, -10, 7, -4, 8, -3, 9, -6, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 68]]
Out[5]=	DTCode[4, 8, 16, 14, 2, 18, 20, 22, 10, 12, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 68]][t]
Out[6]=	-4 6 14 20 2 3 4 -21 - t + -- - -- + -- + 20 t - 14 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 68]][z]
Out[7]=	2 4 6 8 1 + 2 z + 2 z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 68]}
In[9]:=	{KnotDet[Knot[11, Alternating, 68]], KnotSignature[Knot[11, Alternating, 68]]}
Out[9]=	{103, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 68]][q]
Out[10]=	-4 3 6 10 2 3 4 5 6 7 -13 - q + -- - -- + -- + 16 q - 16 q + 15 q - 11 q + 7 q - 4 q + q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 68], Knot[11, Alternating, 111]}
In[12]:=	A2Invariant[Knot[11, Alternating, 68]][q]
Out[12]=	-12 -8 -6 2 2 2 4 6 8 10 12 1 - q + q - q + -- - -- + 2 q - q + 5 q - 2 q + 2 q - q - 4 2 q q 14 16 18 20 > 2 q + q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 68]][a, z]
Out[13]=	2 2 4 4 -4 -2 2 2 z 4 z 2 2 4 3 z 8 z 2 4 2 - a + a - a + 8 z + -- - ---- - 3 a z + 8 z + ---- - ---- - a z + 4 2 4 2 a a a a 6 6 8 6 z 5 z z > 2 z + -- - ---- - -- 4 2 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 68]][a, z]
Out[14]=	2 2 2 -4 -2 2 2 z 3 z z 3 2 z 2 z 7 z 2 - a - a + a + --- + --- + - - a z - a z - 15 z + -- + ---- - ---- - 5 3 a 6 4 2 a a a a a 3 3 3 3 4 2 2 3 z 3 z 4 z 10 z 3 3 3 4 z > 7 a z - ---- - ---- - ---- - ----- - 2 a z + 4 a z + 34 z + -- - 7 5 3 a 8 a a a a 4 4 4 5 5 5 5 7 z 2 z 30 z 2 4 4 z 7 z 5 z 32 z 5 > ---- + ---- + ----- + 14 a z + ---- - ---- + ---- + ----- + 12 a z - 6 4 2 7 5 3 a a a a a a a 6 6 6 7 7 7 3 5 6 7 z 11 z 30 z 2 6 8 z 10 z 33 z > 4 a z - 24 z + ---- - ----- - ----- - 12 a z + ---- - ----- - ----- - 6 4 2 5 3 a a a a a a 8 8 9 9 7 3 7 8 8 z 6 z 2 8 6 z 10 z 9 > 14 a z + a z + z + ---- + ---- + 3 a z + ---- + ----- + 4 a z + 4 2 3 a a a a 10 10 2 z > 2 z + ----- 2 a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 68]], Vassiliev[3][Knot[11, Alternating, 68]]}
Out[15]=	{2, 1}
In[16]:=	Kh[Knot[11, Alternating, 68]][q, t]
Out[16]=	3 1 2 1 4 2 6 4 7 6 q 9 q + 8 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t q t q t 3 5 5 2 7 2 7 3 9 3 9 4 > 8 q t + 8 q t + 7 q t + 8 q t + 4 q t + 7 q t + 3 q t + 11 4 11 5 13 5 15 6 > 4 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a68

K11a67 K11a69