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

K11n85 K11n87

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X12,6,13,5 X7,15,8,14 X9,16,10,17 X2,11,3,12 X13,19,14,18 X15,20,16,21 X17,1,18,22 X19,6,20,7 X21,9,22,8

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 4t-2 - 7t-1 + 9 - 7t + 4t2 - t3

Conway Polynomial:

1 - 2z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{33, 0}

Jones Polynomial:

q-2 - 3q-1 + 5 - 5q + 6q2 - 5q3 + 4q4 - 3q5 + q6

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

{...}

A2 (sl(3)) Invariant:

q-6 - q-4 + q-2 + 2q4 + 2q8 - q10 - q12 - q16 + q18

HOMFLY-PT Polynomial:

2a-4z2 + a-4z4 - 4a-2z2 - 4a-2z4 - a-2z6 + 1 + 2z2 + z4

Kauffman Polynomial:

a-6z2 - 3a-6z4 + a-6z6 - a-5z + 8a-5z3 - 11a-5z5 + 3a-5z7 - 3a-4z2 + 9a-4z4 - 11a-4z6 + 3a-4z8 - 3a-3z + 13a-3z3 - 11a-3z5 + a-3z9 - 9a-2z2 + 22a-2z4 - 17a-2z6 + 4a-2z8 - 3a-1z + 7a-1z3 - 3a-1z7 + a-1z9 + 1 - 5z2 + 10z4 - 5z6 + z8 - az + 2az3

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

{0, 0}

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=0 is the signature of 1186. 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 j = 13                 1 j = 11               2   j = 9             2 1   j = 7           3 2     j = 5         3 2       j = 3       2 3         j = 1     3 3           j = -1   1 3             j = -3   2               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, 86]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 86]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[7, 15, 8, 14], > X[9, 16, 10, 17], X[2, 11, 3, 12], X[13, 19, 14, 18], X[15, 20, 16, 21], > X[17, 1, 18, 22], X[19, 6, 20, 7], X[21, 9, 22, 8]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 86]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, 10, -4, 11, -5, -2, 6, -3, -7, 4, -8, 5, -9, 7, -10, > 8, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 86]]
Out[5]=	DTCode[4, 10, 12, -14, -16, 2, -18, -20, -22, -6, -8]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 86]][t]
Out[6]=	-3 4 7 2 3 9 - t + -- - - - 7 t + 4 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 86]][z]
Out[7]=	4 6 1 - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 86]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 86]], KnotSignature[Knot[11, NonAlternating, 86]]}
Out[9]=	{33, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 86]][q]
Out[10]=	-2 3 2 3 4 5 6 5 + q - - - 5 q + 6 q - 5 q + 4 q - 3 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 86]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 86]][q]
Out[12]=	-6 -4 -2 4 8 10 12 16 18 q - q + q + 2 q + 2 q - q - q - q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 86]][a, z]
Out[13]=	2 2 4 4 6 2 2 z 4 z 4 z 4 z z 1 + 2 z + ---- - ---- + z + -- - ---- - -- 4 2 4 2 2 a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 86]][a, z]
Out[14]=	2 2 2 3 3 3 z 3 z 3 z 2 z 3 z 9 z 8 z 13 z 7 z 1 - -- - --- - --- - a z - 5 z + -- - ---- - ---- + ---- + ----- + ---- + 5 3 a 6 4 2 5 3 a a a a a a a a 4 4 4 5 5 6 6 3 4 3 z 9 z 22 z 11 z 11 z 6 z 11 z > 2 a z + 10 z - ---- + ---- + ----- - ----- - ----- - 5 z + -- - ----- - 6 4 2 5 3 6 4 a a a a a a a 6 7 7 8 8 9 9 17 z 3 z 3 z 8 3 z 4 z z z > ----- + ---- - ---- + z + ---- + ---- + -- + -- 2 5 a 4 2 3 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 86]], Vassiliev[3][Knot[11, NonAlternating, 86]]}
Out[15]=	{0, 0}
In[16]:=	Kh[Knot[11, NonAlternating, 86]][q, t]
Out[16]=	3 1 2 1 3 3 2 5 2 5 3 - + 3 q + ----- + ---- + --- + 3 q t + 2 q t + 3 q t + 3 q t + 2 q t + q 5 2 3 q t q t q t 7 3 7 4 9 4 9 5 11 5 13 6 > 3 q t + 2 q t + 2 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n86

K11n85 K11n87