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

K11n184 K11a1

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X18,4,19,3 X16,5,17,6 X14,8,15,7 X9,21,10,20 X4,12,5,11 X2,13,3,14 X22,16,1,15 X12,18,13,17 X19,9,20,8 X21,11,22,10

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- 2t-3 + 11t-2 - 24t-1 + 31 - 24t + 11t2 - 2t3

Conway Polynomial:

1 + 2z2 - z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{10122, ...}

Determinant and Signature:

{105, 4}

Jones Polynomial:

3q2 - 7q3 + 13q4 - 16q5 + 18q6 - 18q7 + 14q8 - 10q9 + 5q10 - q11

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

{...}

A2 (sl(3)) Invariant:

3q6 - 2q8 + 4q10 + q12 - 2q14 + 4q16 - 4q18 + 2q20 - 3q22 - 2q24 + 2q26 - 3q28 + 3q30 + q32 - q34

HOMFLY-PT Polynomial:

- a-10z2 + a-8 + 5a-8z2 + 3a-8z4 - 4a-6 - 9a-6z2 - 7a-6z4 - 2a-6z6 + 4a-4 + 7a-4z2 + 3a-4z4

Kauffman Polynomial:

a-13z5 - 5a-12z4 + 5a-12z6 + a-11z + 5a-11z3 - 16a-11z5 + 10a-11z7 + 2a-10z2 - 5a-10z4 - 9a-10z6 + 9a-10z8 + a-9z + 16a-9z3 - 35a-9z5 + 14a-9z7 + 3a-9z9 + a-8 - 3a-8z2 + 10a-8z4 - 24a-8z6 + 15a-8z8 - 3a-7z + 12a-7z3 - 18a-7z5 + 7a-7z7 + 3a-7z9 + 4a-6 - 14a-6z2 + 16a-6z4 - 10a-6z6 + 6a-6z8 - 3a-5z + a-5z3 + 3a-5z7 + 4a-4 - 9a-4z2 + 6a-4z4

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=4 is the signature of 11185. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 23                   1 j = 21                 4   j = 19               6 1   j = 17             8 4     j = 15           10 6       j = 13         8 8         j = 11       8 10           j = 9     5 8             j = 7   2 8               j = 5 1 5                 j = 3 3

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, 185]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 185]]
Out[3]=	PD[X[6, 2, 7, 1], X[18, 4, 19, 3], X[16, 5, 17, 6], X[14, 8, 15, 7], > X[9, 21, 10, 20], X[4, 12, 5, 11], X[2, 13, 3, 14], X[22, 16, 1, 15], > X[12, 18, 13, 17], X[19, 9, 20, 8], X[21, 11, 22, 10]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 185]]
Out[4]=	GaussCode[1, -7, 2, -6, 3, -1, 4, 10, -5, 11, 6, -9, 7, -4, 8, -3, 9, -2, -10, > 5, -11, -8]
In[5]:=	DTCode[Knot[11, NonAlternating, 185]]
Out[5]=	DTCode[6, 18, 16, 14, -20, 4, 2, 22, 12, -8, -10]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 185]][t]
Out[6]=	2 11 24 2 3 31 - -- + -- - -- - 24 t + 11 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 185]][z]
Out[7]=	2 4 6 1 + 2 z - z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 122], Knot[11, NonAlternating, 185]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 185]], KnotSignature[Knot[11, NonAlternating, 185]]}
Out[9]=	{105, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 185]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 3 q - 7 q + 13 q - 16 q + 18 q - 18 q + 14 q - 10 q + 5 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 185]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 185]][q]
Out[12]=	6 8 10 12 14 16 18 20 22 24 3 q - 2 q + 4 q + q - 2 q + 4 q - 4 q + 2 q - 3 q - 2 q + 26 28 30 32 34 > 2 q - 3 q + 3 q + q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 185]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -8 4 4 z 5 z 9 z 7 z 3 z 7 z 3 z 2 z a - -- + -- - --- + ---- - ---- + ---- + ---- - ---- + ---- - ---- 6 4 10 8 6 4 8 6 4 6 a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 185]][a, z]
Out[14]=	2 2 2 2 3 -8 4 4 z z 3 z 3 z 2 z 3 z 14 z 9 z 5 z a + -- + -- + --- + -- - --- - --- + ---- - ---- - ----- - ---- + ---- + 6 4 11 9 7 5 10 8 6 4 11 a a a a a a a a a a a 3 3 3 4 4 4 4 4 5 5 16 z 12 z z 5 z 5 z 10 z 16 z 6 z z 16 z > ----- + ----- + -- - ---- - ---- + ----- + ----- + ---- + --- - ----- - 9 7 5 12 10 8 6 4 13 11 a a a a a a a a a a 5 5 6 6 6 6 7 7 7 7 35 z 18 z 5 z 9 z 24 z 10 z 10 z 14 z 7 z 3 z > ----- - ----- + ---- - ---- - ----- - ----- + ----- + ----- + ---- + ---- + 9 7 12 10 8 6 11 9 7 5 a a a a a a a a a a 8 8 8 9 9 9 z 15 z 6 z 3 z 3 z > ---- + ----- + ---- + ---- + ---- 10 8 6 9 7 a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 185]], Vassiliev[3][Knot[11, NonAlternating, 185]]}
Out[15]=	{2, 2}
In[16]:=	Kh[Knot[11, NonAlternating, 185]][q, t]
Out[16]=	3 5 5 7 7 2 9 2 9 3 11 3 3 q + q + 5 q t + 2 q t + 8 q t + 5 q t + 8 q t + 8 q t + 11 4 13 4 13 5 15 5 15 6 17 6 > 10 q t + 8 q t + 8 q t + 10 q t + 6 q t + 8 q t + 17 7 19 7 19 8 21 8 23 9 > 4 q t + 6 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n185

K11n184 K11a1