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

K11a69 K11a71

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 18 12 2 16 20 6 10 22 14

Alexander Polynomial:

2t-3 - 14t-2 + 36t-1 - 47 + 36t - 14t2 + 2t3

Conway Polynomial:

1 - 2z2 - 2z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{151, 2}

Jones Polynomial:

q-3 - 3q-2 + 8q-1 - 14 + 20q - 24q2 + 25q3 - 22q4 + 17q5 - 11q6 + 5q7 - q8

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

{...}

A2 (sl(3)) Invariant:

q-10 - q-6 + 4q-4 - 2q-2 - 1 + 4q2 - 5q4 + 3q6 - 2q8 + 3q12 - 4q14 + 5q16 - 2q18 - 2q20 + 3q22 - q24

HOMFLY-PT Polynomial:

- a-6z4 + a-4 + a-4z2 + a-4z4 + a-4z6 - a-2 - 2a-2z2 + a-2z6 - 2z2 - 2z4 + a2 + a2z2

Kauffman Polynomial:

a-9z5 - 4a-8z4 + 5a-8z6 + 4a-7z3 - 15a-7z5 + 11a-7z7 + 5a-6z4 - 18a-6z6 + 13a-6z8 - a-5z + 12a-5z3 - 22a-5z5 + 2a-5z7 + 8a-5z9 + a-4 - 7a-4z2 + 30a-4z4 - 46a-4z6 + 20a-4z8 + 2a-4z10 - a-3z + 8a-3z3 - 4a-3z5 - 16a-3z7 + 13a-3z9 + a-2 - 10a-2z2 + 28a-2z4 - 33a-2z6 + 12a-2z8 + 2a-2z10 - a-1z + 5a-1z3 - 5a-1z5 - 4a-1z7 + 5a-1z9 + 4z4 - 9z6 + 5z8 - az + 5az3 - 7az5 + 3az7 - a2 + 3a2z2 - 3a2z4 + a2z6

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 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 r = 5 r = 6 r = 7 j = 17                       1 j = 15                     4   j = 13                   7 1   j = 11                 10 4     j = 9               12 7       j = 7             13 10         j = 5           11 12           j = 3         9 13             j = 1       6 12               j = -1     2 8                 j = -3   1 6                   j = -5   2                     j = -7 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, 70]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 70]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[18, 6, 19, 5], X[12, 8, 13, 7], > X[2, 9, 3, 10], X[16, 11, 17, 12], X[20, 14, 21, 13], X[6, 16, 7, 15], > X[10, 17, 11, 18], X[22, 20, 1, 19], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 70]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -8, 4, -2, 5, -9, 6, -4, 7, -11, 8, -6, 9, -3, 10, > -7, 11, -10]
In[5]:=	DTCode[Knot[11, Alternating, 70]]
Out[5]=	DTCode[4, 8, 18, 12, 2, 16, 20, 6, 10, 22, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 70]][t]
Out[6]=	2 14 36 2 3 -47 + -- - -- + -- + 36 t - 14 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 70]][z]
Out[7]=	2 4 6 1 - 2 z - 2 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 70]}
In[9]:=	{KnotDet[Knot[11, Alternating, 70]], KnotSignature[Knot[11, Alternating, 70]]}
Out[9]=	{151, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 70]][q]
Out[10]=	-3 3 8 2 3 4 5 6 7 8 -14 + q - -- + - + 20 q - 24 q + 25 q - 22 q + 17 q - 11 q + 5 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 70]}
In[12]:=	A2Invariant[Knot[11, Alternating, 70]][q]
Out[12]=	-10 -6 4 2 2 4 6 8 12 14 16 -1 + q - q + -- - -- + 4 q - 5 q + 3 q - 2 q + 3 q - 4 q + 5 q - 4 2 q q 18 20 22 24 > 2 q - 2 q + 3 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 70]][a, z]
Out[13]=	2 2 4 4 6 6 -4 -2 2 2 z 2 z 2 2 4 z z z z a - a + a - 2 z + -- - ---- + a z - 2 z - -- + -- + -- + -- 4 2 6 4 4 2 a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 70]][a, z]
Out[14]=	2 2 3 3 -4 -2 2 z z z 7 z 10 z 2 2 4 z 12 z a + a - a - -- - -- - - - a z - ---- - ----- + 3 a z + ---- + ----- + 5 3 a 4 2 7 5 a a a a a a 3 3 4 4 4 4 5 8 z 5 z 3 4 4 z 5 z 30 z 28 z 2 4 z > ---- + ---- + 5 a z + 4 z - ---- + ---- + ----- + ----- - 3 a z + -- - 3 a 8 6 4 2 9 a a a a a a 5 5 5 5 6 6 6 15 z 22 z 4 z 5 z 5 6 5 z 18 z 46 z > ----- - ----- - ---- - ---- - 7 a z - 9 z + ---- - ----- - ----- - 7 5 3 a 8 6 4 a a a a a a 6 7 7 7 7 8 33 z 2 6 11 z 2 z 16 z 4 z 7 8 13 z > ----- + a z + ----- + ---- - ----- - ---- + 3 a z + 5 z + ----- + 2 7 5 3 a 6 a a a a a 8 8 9 9 9 10 10 20 z 12 z 8 z 13 z 5 z 2 z 2 z > ----- + ----- + ---- + ----- + ---- + ----- + ----- 4 2 5 3 a 4 2 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 70]], Vassiliev[3][Knot[11, Alternating, 70]]}
Out[15]=	{-2, -1}
In[16]:=	Kh[Knot[11, Alternating, 70]][q, t]
Out[16]=	3 1 2 1 6 2 8 6 q 3 12 q + 9 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 13 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 11 q t + 12 q t + 13 q t + 10 q t + 12 q t + 7 q t + 10 q t + 11 5 13 5 13 6 15 6 17 7 > 4 q t + 7 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a70

K11a69 K11a71