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

K11n183 K11n185

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DrawMorseLink

PD Presentation:

X6271 X3,15,4,14 X10,6,11,5 X18,7,19,8 X2,10,3,9 X22,11,1,12 X20,14,21,13 X15,5,16,4 X12,18,13,17 X8,19,9,20 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

2t-3 - 9t-2 + 20t-1 - 25 + 20t - 9t2 + 2t3

Conway Polynomial:

1 + 2z2 + 3z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{1084, K11a46, ...}

Determinant and Signature:

{87, 2}

Jones Polynomial:

- 2 + 7q - 10q2 + 14q3 - 15q4 + 14q5 - 12q6 + 8q7 - 4q8 + q9

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

{...}

A2 (sl(3)) Invariant:

- 2 + 3q2 - q4 + 2q6 + 4q8 - 2q10 + 3q12 - 3q14 + q16 - 3q20 + 2q22 - 2q24 + q28

HOMFLY-PT Polynomial:

a-8 + a-8z2 - 4a-6 - 7a-6z2 - 3a-6z4 + 4a-4 + 11a-4z2 + 8a-4z4 + 2a-4z6 - 3a-2z2 - 2a-2z4

Kauffman Polynomial:

a-10z2 - 2a-10z4 + a-10z6 - 2a-9z + 7a-9z3 - 10a-9z5 + 4a-9z7 + a-8 - a-8z2 + 6a-8z4 - 14a-8z6 + 6a-8z8 - 7a-7z + 24a-7z3 - 28a-7z5 + 4a-7z7 + 3a-7z9 + 4a-6 - 14a-6z2 + 29a-6z4 - 36a-6z6 + 14a-6z8 - 7a-5z + 25a-5z3 - 27a-5z5 + 6a-5z7 + 3a-5z9 + 4a-4 - 19a-4z2 + 29a-4z4 - 20a-4z6 + 8a-4z8 - 3a-3z + 11a-3z3 - 9a-3z5 + 6a-3z7 - 7a-2z2 + 8a-2z4 + a-2z6 - a-1z + 3a-1z3

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

{2, 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=2 is the signature of 11184. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 19                   1 j = 17                 3   j = 15               5 1   j = 13             7 3     j = 11           7 5       j = 9         8 7         j = 7       6 7           j = 5     4 8             j = 3   3 6               j = 1   5                 j = -1 2

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, 184]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 184]]
Out[3]=	PD[X[6, 2, 7, 1], X[3, 15, 4, 14], X[10, 6, 11, 5], X[18, 7, 19, 8], > X[2, 10, 3, 9], X[22, 11, 1, 12], X[20, 14, 21, 13], X[15, 5, 16, 4], > X[12, 18, 13, 17], X[8, 19, 9, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 184]]
Out[4]=	GaussCode[1, -5, -2, 8, 3, -1, 4, -10, 5, -3, 6, -9, 7, 2, -8, -11, 9, -4, 10, > -7, 11, -6]
In[5]:=	DTCode[Knot[11, NonAlternating, 184]]
Out[5]=	DTCode[6, -14, 10, 18, 2, 22, 20, -4, 12, 8, 16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 184]][t]
Out[6]=	2 9 20 2 3 -25 + -- - -- + -- + 20 t - 9 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 184]][z]
Out[7]=	2 4 6 1 + 2 z + 3 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 84], Knot[11, Alternating, 46], Knot[11, NonAlternating, 184]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 184]], KnotSignature[Knot[11, NonAlternating, 184]]}
Out[9]=	{87, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 184]][q]
Out[10]=	2 3 4 5 6 7 8 9 -2 + 7 q - 10 q + 14 q - 15 q + 14 q - 12 q + 8 q - 4 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 184]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 184]][q]
Out[12]=	2 4 6 8 10 12 14 16 20 22 -2 + 3 q - q + 2 q + 4 q - 2 q + 3 q - 3 q + q - 3 q + 2 q - 24 28 > 2 q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 184]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -8 4 4 z 7 z 11 z 3 z 3 z 8 z 2 z 2 z a - -- + -- + -- - ---- + ----- - ---- - ---- + ---- - ---- + ---- 6 4 8 6 4 2 6 4 2 4 a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 184]][a, z]
Out[14]=	2 2 2 2 2 -8 4 4 2 z 7 z 7 z 3 z z z z 14 z 19 z 7 z a + -- + -- - --- - --- - --- - --- - - + --- - -- - ----- - ----- - ---- + 6 4 9 7 5 3 a 10 8 6 4 2 a a a a a a a a a a a 3 3 3 3 3 4 4 4 4 4 7 z 24 z 25 z 11 z 3 z 2 z 6 z 29 z 29 z 8 z > ---- + ----- + ----- + ----- + ---- - ---- + ---- + ----- + ----- + ---- - 9 7 5 3 a 10 8 6 4 2 a a a a a a a a a 5 5 5 5 6 6 6 6 6 7 10 z 28 z 27 z 9 z z 14 z 36 z 20 z z 4 z > ----- - ----- - ----- - ---- + --- - ----- - ----- - ----- + -- + ---- + 9 7 5 3 10 8 6 4 2 9 a a a a a a a a a a 7 7 7 8 8 8 9 9 4 z 6 z 6 z 6 z 14 z 8 z 3 z 3 z > ---- + ---- + ---- + ---- + ----- + ---- + ---- + ---- 7 5 3 8 6 4 7 5 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 184]], Vassiliev[3][Knot[11, NonAlternating, 184]]}
Out[15]=	{2, 2}
In[16]:=	Kh[Knot[11, NonAlternating, 184]][q, t]
Out[16]=	3 2 3 5 5 2 7 2 7 3 9 3 5 q + 3 q + --- + 6 q t + 4 q t + 8 q t + 6 q t + 7 q t + 8 q t + q t 9 4 11 4 11 5 13 5 13 6 15 6 15 7 > 7 q t + 7 q t + 5 q t + 7 q t + 3 q t + 5 q t + q t + 17 7 19 8 > 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n184

K11n183 K11n185