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

K11n80 K11n82

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X5,16,6,17 X2837 X9,21,10,20 X11,1,12,22 X13,19,14,18 X15,6,16,7 X17,13,18,12 X19,15,20,14 X21,11,22,10

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-4 + 3t-3 - 4t-2 + 4t-1 - 3 + 4t - 4t2 + 3t3 - t4

Conway Polynomial:

1 - z2 - 6z4 - 5z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{27, 6}

Jones Polynomial:

q - q2 + 4q3 - 4q4 + 4q5 - 5q6 + 3q7 - 3q8 + 2q9

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

{...}

A2 (sl(3)) Invariant:

q4 + q6 + 3q8 + 3q10 + 2q12 - 4q16 - 3q18 - 5q20 + q24 + 2q26 + 2q28 + q32 - q34

HOMFLY-PT Polynomial:

- a-10 - a-10z2 + 7a-8 + 11a-8z2 + 6a-8z4 + a-8z6 - 13a-6 - 23a-6z2 - 18a-6z4 - 7a-6z6 - a-6z8 + 8a-4 + 12a-4z2 + 6a-4z4 + a-4z6

Kauffman Polynomial:

- 3a-11z + a-11z3 + a-10 + 8a-10z2 - 12a-10z4 + 3a-10z6 - 18a-9z + 41a-9z3 - 30a-9z5 + 6a-9z7 + 7a-8 - 10a-8z2 + 20a-8z4 - 18a-8z6 + 4a-8z8 - 26a-7z + 53a-7z3 - 30a-7z5 + 2a-7z7 + a-7z9 + 13a-6 - 38a-6z2 + 50a-6z4 - 28a-6z6 + 5a-6z8 - 11a-5z + 13a-5z3 - 4a-5z7 + a-5z9 + 8a-4 - 20a-4z2 + 18a-4z4 - 7a-4z6 + a-4z8

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

{-1, -6}

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=6 is the signature of 1181. 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 = 19                 2 j = 17               1   j = 15             3 3   j = 13           3 1     j = 11         1 3 1     j = 9       3 3         j = 7     1 1           j = 5   1 4             j = 3                   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, 81]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 81]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 16, 6, 17], X[2, 8, 3, 7], > X[9, 21, 10, 20], X[11, 1, 12, 22], X[13, 19, 14, 18], X[15, 6, 16, 7], > X[17, 13, 18, 12], X[19, 15, 20, 14], X[21, 11, 22, 10]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 81]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 8, 4, -2, -5, 11, -6, 9, -7, 10, -8, 3, -9, 7, -10, > 5, -11, 6]
In[5]:=	DTCode[Knot[11, NonAlternating, 81]]
Out[5]=	DTCode[4, 8, -16, 2, -20, -22, -18, -6, -12, -14, -10]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 81]][t]
Out[6]=	-4 3 4 4 2 3 4 -3 - t + -- - -- + - + 4 t - 4 t + 3 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 81]][z]
Out[7]=	2 4 6 8 1 - z - 6 z - 5 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 81]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 81]], KnotSignature[Knot[11, NonAlternating, 81]]}
Out[9]=	{27, 6}
In[10]:=	J=Jones[Knot[11, NonAlternating, 81]][q]
Out[10]=	2 3 4 5 6 7 8 9 q - q + 4 q - 4 q + 4 q - 5 q + 3 q - 3 q + 2 q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 81]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 81]][q]
Out[12]=	4 6 8 10 12 16 18 20 24 26 28 q + q + 3 q + 3 q + 2 q - 4 q - 3 q - 5 q + q + 2 q + 2 q + 32 34 > q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 81]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -10 7 13 8 z 11 z 23 z 12 z 6 z 18 z 6 z z -a + -- - -- + -- - --- + ----- - ----- + ----- + ---- - ----- + ---- + -- - 8 6 4 10 8 6 4 8 6 4 8 a a a a a a a a a a a 6 6 8 7 z z z > ---- + -- - -- 6 4 6 a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 81]][a, z]
Out[14]=	2 2 2 2 -10 7 13 8 3 z 18 z 26 z 11 z 8 z 10 z 38 z 20 z a + -- + -- + -- - --- - ---- - ---- - ---- + ---- - ----- - ----- - ----- + 8 6 4 11 9 7 5 10 8 6 4 a a a a a a a a a a a 3 3 3 3 4 4 4 4 5 z 41 z 53 z 13 z 12 z 20 z 50 z 18 z 30 z > --- + ----- + ----- + ----- - ----- + ----- + ----- + ----- - ----- - 11 9 7 5 10 8 6 4 9 a a a a a a a a a 5 6 6 6 6 7 7 7 8 8 30 z 3 z 18 z 28 z 7 z 6 z 2 z 4 z 4 z 5 z > ----- + ---- - ----- - ----- - ---- + ---- + ---- - ---- + ---- + ---- + 7 10 8 6 4 9 7 5 8 6 a a a a a a a a a a 8 9 9 z z z > -- + -- + -- 4 7 5 a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 81]], Vassiliev[3][Knot[11, NonAlternating, 81]]}
Out[15]=	{-1, -6}
In[16]:=	Kh[Knot[11, NonAlternating, 81]][q, t]
Out[16]=	5 5 7 q q 7 9 9 2 11 2 11 3 13 3 4 q + q + -- + -- + q t + 3 q t + 3 q t + q t + 3 q t + 3 q t + 2 t t 11 4 13 4 15 4 15 5 17 5 19 6 > q t + q t + 3 q t + 3 q t + q t + 2 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n81

K11n80 K11n82