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

K11n182 K11n184

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X3,15,4,14 X10,6,11,5 X7,19,8,18 X2,10,3,9 X22,11,1,12 X20,14,21,13 X15,5,16,4 X12,18,13,17 X19,9,20,8 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:

t-3 + t-2 - 6t-1 + 9 - 6t + t2 + t3

Conway Polynomial:

1 + 7z2 + 7z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{21, 4}

Jones Polynomial:

q3 - q5 + 3q6 - 3q7 + 4q8 - 4q9 + 3q10 - 3q11 + q12

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

{...}

A2 (sl(3)) Invariant:

q10 + q12 + q16 + q18 + 3q22 + q24 + q26 - 2q28 - 3q30 - q32 - 2q34 + q36 + q38

HOMFLY-PT Polynomial:

a-12 - 4a-10 - 3a-10z2 + 2a-8 + 2a-8z2 + a-8z4 + 2a-6 + 8a-6z2 + 6a-6z4 + a-6z6

Kauffman Polynomial:

a-14z2 - 3a-14z4 + a-14z6 - a-13z + 10a-13z3 - 12a-13z5 + 3a-13z7 + a-12 - a-12z2 + 4a-12z4 - 7a-12z6 + 2a-12z8 - 9a-11z + 26a-11z3 - 22a-11z5 + 5a-11z7 + 4a-10 - 14a-10z2 + 17a-10z4 - 10a-10z6 + 2a-10z8 - 8a-9z + 18a-9z3 - 11a-9z5 + 2a-9z7 + 2a-8 - 4a-8z2 + 4a-8z4 - a-8z6 + 2a-7z3 - a-7z5 - 2a-6 + 8a-6z2 - 6a-6z4 + a-6z6

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

{7, 17}

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 11183. 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 r = 10 j = 25                     1 j = 23                   2   j = 21                 1 1   j = 19               3 2     j = 17           1 2 1       j = 15           2 3         j = 13       1 3 2           j = 11         2             j = 9     1 2               j = 7 1                     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, 183]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 183]]
Out[3]=	PD[X[6, 2, 7, 1], X[3, 15, 4, 14], X[10, 6, 11, 5], X[7, 19, 8, 18], > 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[19, 9, 20, 8], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 183]]
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, 183]]
Out[5]=	DTCode[6, -14, 10, -18, 2, 22, 20, -4, 12, -8, 16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 183]][t]
Out[6]=	-3 -2 6 2 3 9 + t + t - - - 6 t + t + t t
In[7]:=	Conway[Knot[11, NonAlternating, 183]][z]
Out[7]=	2 4 6 1 + 7 z + 7 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 183]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 183]], KnotSignature[Knot[11, NonAlternating, 183]]}
Out[9]=	{21, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 183]][q]
Out[10]=	3 5 6 7 8 9 10 11 12 q - q + 3 q - 3 q + 4 q - 4 q + 3 q - 3 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 183]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 183]][q]
Out[12]=	10 12 16 18 22 24 26 28 30 32 34 36 q + q + q + q + 3 q + q + q - 2 q - 3 q - q - 2 q + q + 38 > q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 183]][a, z]
Out[13]=	2 2 2 4 4 6 -12 4 2 2 3 z 2 z 8 z z 6 z z a - --- + -- + -- - ---- + ---- + ---- + -- + ---- + -- 10 8 6 10 8 6 8 6 6 a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 183]][a, z]
Out[14]=	2 2 2 2 2 -12 4 2 2 z 9 z 8 z z z 14 z 4 z 8 z a + --- + -- - -- - --- - --- - --- + --- - --- - ----- - ---- + ---- + 10 8 6 13 11 9 14 12 10 8 6 a a a a a a a a a a a 3 3 3 3 4 4 4 4 4 5 10 z 26 z 18 z 2 z 3 z 4 z 17 z 4 z 6 z 12 z > ----- + ----- + ----- + ---- - ---- + ---- + ----- + ---- - ---- - ----- - 13 11 9 7 14 12 10 8 6 13 a a a a a a a a a a 5 5 5 6 6 6 6 6 7 7 7 22 z 11 z z z 7 z 10 z z z 3 z 5 z 2 z > ----- - ----- - -- + --- - ---- - ----- - -- + -- + ---- + ---- + ---- + 11 9 7 14 12 10 8 6 13 11 9 a a a a a a a a a a a 8 8 2 z 2 z > ---- + ---- 12 10 a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 183]], Vassiliev[3][Knot[11, NonAlternating, 183]]}
Out[15]=	{7, 17}
In[16]:=	Kh[Knot[11, NonAlternating, 183]][q, t]
Out[16]=	5 7 9 2 9 3 13 3 11 4 13 4 13 5 q + q + q t + 2 q t + q t + 2 q t + 3 q t + 2 q t + 15 5 17 5 15 6 17 6 17 7 19 7 19 8 > 2 q t + q t + 3 q t + 2 q t + q t + 3 q t + 2 q t + 21 8 21 9 23 9 25 10 > q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n183

K11n182 K11n184